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\begin{document}
\title{Editorial Commentary on the Lie-Bianchi Classification of 3-Dimensional Lie Algebras}
\author{Robert T. Jantzen}
\date{}
\maketitle
\begin{abstract}
The Bianchi classification of 3-dimensional Lie algebras, a refinement of Lie's original work from the complex to the real case, has played an important role in spatially homogeneous cosmology, a topic in relativity that has itself been a fruitful arena for testing many ideas in gravitational theory. The translation from the original Italian to English of the derivation of this classification taken from Bianchi's own book on Lie Groups is given here with some editorial commentary placing it in context with the classification scheme actually used in practice as introduced by Sch\"ucking.
\end{abstract}
The journal {\it General Relativity and Gravitation\/}, in its Golden Oldies series \cite{kra} of reprints of historically important articles which are relatively inaccessible or not written in English or both, accompanied by editorial commentary placing them into context, has made available key documents (in English translation if not originally in English) associated with the development of spatially homogeneous relativistic cosmology. Chronologically these begin with the English translation of Luigi Bianchi's 1898 local classification of isometry classes of 3-dimensional Riemannian manifolds \cite{bianchipaper}, representative metrics of which provide the 3-geometries whose time evolution in relativistic gravitational theories describe spatially homogeneous cosmological models. These were first introduced into relativistic cosmology by Kurt G\"odel in 1949 \cite{godel1} and 1951 \cite{godel2}, then systematically for empty universes by Abraham Taub in 1951 \cite{taub}, after which various others began investigating first dust-filled and then more general perfect-fluid filled models in a field of work that peaked in the 1970s \cite{ryan} when I entered the field as an undergraduate at Princeton and then as Abe Taub's graduate student at U.C. Berkeley just before his retirement in 1978 \cite{jantzen}.
Ellis and MacCallum \cite{ellis} systematized the Lie algebra structure constant decomposition for use in studying spatially homogeneous dynamics, with a final detail added by Hawking and Collins \cite{hawking}.
A brief overview of this modern history can be found in the historical commentary of \cite{bianchipaper}.
The next important document in this story was never written or published as originally intended, but has been recreated \cite{kundt} from the German lecture notes of 1956 taken by Wolfgang Kundt of Engelbert Sch\"ucking's description of an alternative approach to the classification of 3-dimensional Lie algebras for application to dust-filled cosmological models \cite{heckmann}, which indirectly found its way into print in an article coauthored by Sch\"ucking's student Christoph Behr \cite{behr} in 1962. The Lie algebra of the symmetry group for these models is the key ingredient for studying their dynamics and all of their properties, so the algebraic properties of its structure constant tensor (Lie bracket) are crucial.
Lie's fundamental work on transformation groups appeared in three volumes written with Engel's assistance in 1888--1893 \cite{lie1} and another written with Scheffers in 1893 \cite{lie2} (see also \cite{lie3}). He classified all Lie algebras up to dimension 6 over the complex numbers. Killing used this work to analyze the symmetries of Riemannian metrics, publishing his famous equations in 1893 whose solutions give the generating vector fields of isometries. A few years later Bianchi used both men's work to study the local isometry classes of 3-dimensional Riemannian metrics, refining Lie's classification of 3-dimensional Lie algebras slightly from the complex to the real case, and
explicitly giving representative generating Lie algebras and metrics for each symmetry type. Later in his own book on continuous groups \cite{bianchibook}, Bianchi presented his version of Lie's derivation of this classification scheme, the English translation of which is the document given below.
The Princeton differential geometer Luther Eisenhart later gave his own accounts of this work in continous groups and differential geometry in English \cite{eisenhart1,eisenhart2}, which must have been responsible for Kurt G\"odel and Abraham Taub's awareness of Bianchi's work. Both participated in the joint activities of the Institute for Advanced Study and the Princeton University mathematics department that Eisenhart chaired during the 1930s, and Taub revisited the Institute in the late 1940s when G\"odel, as Einstein's best friend there, began pursuing his rotating universe obsession. The Princeton mathematics story is available on-line \cite{pmc}, while biographical information about the various mathematicians themselves can be found in Gillispie's Dictionary of Scientific Biography \cite{gillispie} and in the MacTutor History of Mathematics Archive \cite{mactutor}.
Bianchi's derivation of the Lie classification uses the dimension of the derived group or Lie algebra as the main tool, while the Sch\"ucking approach instead focuses on the invariance properties of the Lie algebra's structure constant tensor under linear transformations.
{\bf[in progress]}
\begin{thebibliography}{00}
\bibitem{kra}
Andrzej Krasi\'nski, ``Golden Oldies,"
{\it Gen.\ Rel\. Grav.\/} {\bf 29}, 357--358 (1997).
\bibitem{bianchipaper}
{\it Sugli spazi a tre dimensioni che ammettono un
gruppo continuo di movimenti\/}
(On 3-Dimensional Spaces Which Admit a Continuous Group of Motions),
{\it Memorie di Matematica e di Fisica della
Societa Italiana delle Scienze\/},
Third Series, Vol.~XI (1898), pp.~267--352;
reprinted in:
{\it Opere\/} (The Collected Works of Luigi Bianchi),
Rome, Edizione Cremonese, 1952, vol.~9, pp.~17--109;
English translation with editorial comment by Robert T. Jantzen
in {\it Gen.\ Rel.\ Grav.\/} {\bf 33}, 2157--2253 (2001).
\bibitem{godel1}
Kurt G\"odel,
``An Example of a New Type of Cosmological Solutions of Einstein's Field
Equations of Gravitation,"
{\it Reviews of Modern Physics\/} {\bf 21}, 447 (1949); reprinted in
{\it Gen.\ Rel. \ Grav.\/} {\bf 32}, 1409--1417 (2000).
\bibitem{godel2}
Kurt G\"odel,
``Rotating Universes in General Relativity Theory,"
{\it Proceedings of the
International Congress of Mathematicians\/},
Cambridge, Mass. 1950, Vol. 1, 175,
Amer.\ Math.\ Soc., R.I., 1952;
reprinted in {\it Gen.\ Rel. \ Grav.\/} {\bf 32}, 1419--1427 (2000).
\bibitem{taub}
Abraham Taub,
``Empty Spacetimes Admitting a Three-Parameter Group of
Motions,"
{\it Proceedings of the
International Congress of Mathematicians (Cambridge, Mass., 1950),
p.655\/}; {\it Annals of Mathematics\/} {\bf 53}, 472--490 (1951);
reprinted in
{\it Gen.\ Rel.\ Grav.\/} {\bf 34}, (2002).
\bibitem{ryan}
Michael P. Ryan, Jr.\ and Lawrence C. Shepley,
{\it Homogeneous Relativistic Cosmolgies\/},
Princeton University Press, Princeton, 1975.
\bibitem{jantzen}
R.T. Jantzen,
``The Dynamical Degrees of Freedom in Spatially Homogeneous Cosmology,"
{\it Commun.\ Math.\ Phys.\/} {\bf 64}, 211--232 (1978).
\bibitem{ellis}
G.F.R. Ellis and M.A.H. MacCallum,
``A Class of Homogeneous Cosmological
Models," {\it Commun.\ Math.\ Phys.\/} {\bf 12}, 108--141 (1969).
\bibitem{hawking}
C. B. Collins and S. W. Hawking,
``Why is the Universe Isotropic?,"
{\it Ap.\ J.\/} {\bf 180}, 317--334 (1973).
\bibitem{kundt}
W. Kundt,
``The Bianchi Classification in the Sch\"ucking-Behr Approach,"
{\it Gen.\ Rel. \ Grav.\/} {\bf 34}, to appear (2002).
\bibitem{heckmann}
O. Heckmann and E. Sch\"ucking,
``Relativistic Cosmology,"
in {\it Gravitation, an Introduction to
Current Research\/}, edited by Louis Witten, Wiley, New York, 1962, pp.~438--469.
\bibitem{behr}
F.B. Estabrook, H.D. Wahlquist, and C.G. Behr,
``Dyadic Analysis of Spatially Homogeneous World Models,"
{\it J.\ Math.\ Phys.\/} {\bf 9}, 497--504 (1968).
\bibitem{lie1}
Sophus Lie (with Friedrich Engel), {\it Theorie der Transformationsgruppen\/}
(Theory of Transformation Groups), Volumes 1--3, Liepzig, Germany,
1888, 1890, 1893 [reprinted by Chelsea Pub Co, Bronx, NY 1970],
see chapter 28, section 136, pp.~713--722.
\bibitem{lie2}
Sophus Lie and Georg Scheffers,
{\it Vorlesungen \"uber Continuierliche Gruppen
mit Geometrischen und Anderen Anwendungen\/}
(Lectures on Continuous Groups with Geometric and Other Applications), Leipzig,
Germany, 1983 [reprinted by Chelsea Pub Co, Bronx, NY 1971],
see chapter 20, section 2, pp.~565--572.
\bibitem{lie3}
Lie Groups: History, Frontiers and Applications Volume 1: Sophus Lie's 1880
Transformation Group Paper, translated by Michael Ackermann, comments by Robert
Hermann, Math Sci Press, Brookline, Massachussetts, 1976.
\bibitem{killing}
W. Killing,
``\"Uber die Grundlagen der Geometrie" (On the Foundations of Geometry),
{\it Journ. f\"ur die r. und ang. Math. (Crelle)\/},
{\bf 109}, pp.~121--186 (1892).
\bibitem{bianchibook}
Luigi Bianchi, {\it Lezioni sulla teoria dei gruppi continui finiti di trasformazioni\/}
(Lectures on the theory of
finite continuous transformation groups (1902-1903)),
Spoerri, Pisa, 1918, see sections 198--199, pp.~550--557.
\bibitem{eisenhart1}
Luther P. Eisenhart, {\it Riemannian Geometry\/}, Princeton University Press, 1925.
\bibitem{eisenhart2}
Luther P Eisenhart, {\it Continous Groups of Transformations\/}, Princeton University Press, 1933 (Dover Edition, 1961).
\bibitem{pmc}
{\it The Princeton Mathematics Community in the 1930s: An Oral History
Project\/}, Charles C. Gillispie, Albert W. Tucker, William Aspray, and
Frederik Nebeker, Princeton University, 1985; now on-line at {\tt
http://www.princeton.edu/mudd/math}.
\bibitem{gillispie}
Charles C. Gillispie, editor,
{\it Dictionary of Scientific Biography\/},
Scribner's Sons, New York, 1970.
\bibitem{mactutor}
MacTutor History of Mathematics Archive:\\
{\tt http://www-history.mcs.st-andrews.ac.uk/}
\end{thebibliography}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{
The Bianchi Classification of 3-Dimensional Lie Algebras:\\
Lezioni sulla teoria dei gruppi continui finite di trasformazioni
(1918) \S198--199 (pp.~550--557)}
\author{Luigi Bianchi\\
\small English translation by Robert T. Jantzen, May 28, 1999}
\date{}
\maketitle
\section*{\S198. The Seven Types of Compositions of Integrable $G_3$'s}
The direction of our investigation leads us to determine all the spaces
$S_3$ which admit a {\it transitive\/}
and therefore simply transitive group $G_3$ of motions.
And since two such $G_3$'s are always similar when equally
composed (\S98), the first classification to make is that of all
possible compositions for a $G_3$; then having chosen a
specific representative for each of these types, we have
to study the spaces which admit the $G_3$ as a (complete or
partial) group of motions. In this and the following
section we occupy ourselves with the preliminary study
of the types of compositions for a $G_3$.
We distinguish the $G_3$'s according to whether they are
integrable or nonintegrable. If a $G_3$ is integrable,
its derived group contains less than 3 parameters (\S81 $a)$),
and conversely if this happens the $G_3$ is integrable
since the derived group, having fewer than 3 parameters,
is certainly integrable.
A $G_3$ is therefore not integrable only when it
coincides with its own derived group, in which case
it is simple.\footnote{%(l)
It cannot possess an invariant $G_2$ (\S81 $a)$), nor even
an invariant $G_1$ since in such a case the derived
group would have at most two parameters.}
We treat in this section the case of an
integrable $G_3$ . Its derived group will fall into
one of the following three categories:
$a)$ it reduces to the identity
$b)$ it is a $G_1 \equiv [X_1 f]$
$c)$ it is a $G_2 \equiv [X_1 f, X_2 f]$.
\subsection*{Case $a)$}
The group $G_3$ is abelian and offers the first and
simplest composition
$$
[X_1, X_2]f =
[X_1, X_3]f =
[X_2, X_3]f = 0\ .
\leqno{\rm Type\ I:}
$$
\subsection*{Case $b)$}
Here we have
$$
[X_1, X_2]f = \alpha X_1 f\ ,\
[X_1, X_3]f = \beta X_1 f\ ,\
[X_2, X_3]f = \gamma X_1 f\ .
$$
with $\alpha,\beta,\gamma$ constants which are not simultaneously zero
(otherwise we would be in case $a)$).
But one of the three, for instance $\alpha$,
can be made zero;
it suffices (when $\alpha\neq0$) to change
$X_2 f $ into $\bar X_2 f= -\beta/\alpha \,X_2 f + X_3 f$.
Therefore we can assume
$$
[X_1, X_2]f = 0\ ,\
[X_1, X_3]f = \beta X_1 f\ ,\
[X_2, X_3]f = \gamma X_1 f\ .
$$
and if also
$\beta=0 $ it suffices to multiply $X_3 f$
by a constant factor to make $\gamma=1 $
leading to
$$
[X_1, X_2]f =
[X_1, X_3]f = 0\ ,\
[X_2, X_3]f = X_1 f\ .
\leqno{\rm Type\ II:}
$$
If then $\beta\neq0$
we can make $\beta=1 $ and render $\gamma=0 $, if it is not
already so, by changing $X_2 f $ into
$\bar X_1 f -1/\gamma\, X_2 f $ so that
$$
[X_1, X_2]f = [X_1, X_3]f -1/\gamma\, [X_2, X_3]f = 0\ .
$$
Therefore we obtain\footnote{%(1)
That the types II, III are actually different
is shown for example by the observation that
the derived group $[X_1 f]f $ is contained in the
first case in $\infty^1 $ abelian $G_2$'s
$[X_1 f, a X_2 f+ b X_3 f]$
but in the second only in the abelian $G_2 $
$[X_1 f, X_2 f]$.}
$$
[X_1, X_2]f = 0\ ,\
[X_1, X_3]f = X_1 f\ ,\
[X_2, X_3]f = 0\ .
\leqno{\rm Type\ III:}
$$
\subsection*{Case $c)$}
We begin by proving that in this case the
derived group $[X_1 f, X_2 f] $
is necessarily abelian.
Suppose on the contrary that
$[X_1, X_2]f = a X_1 f + b X_2 f $
with the constants $a,b $ {\it not both zero\/}, and one has
$$
[X_1, X_3]f = \alpha X_1 f + \beta X_2 f\ ,\
[X_2, X_3]f = \gamma X_1 f + \delta X_2 f\ .
$$
The Jacobi identity
$
[[X_1,X_2],X_3] + [[X_2,X_3],X_1] + [[X_3,X_1],X_2] =0
$
gives among the constants the two relations
$b \gamma - a \delta = 0$, $b \alpha -a \beta =0 $,
so that, $a$ and $b$ not both zero by hypothesis,
we must have $\alpha \delta -\beta \gamma = 0 $.
The matrix
$\pmatrix{a & \alpha & \gamma\cr b & \beta & \delta\cr}$
would therefore have rank 1 and the three linear
forms in $X_1 f$, $X_2 f$,
$$
a X_1 + b X_2\ ,\
\alpha X_1 + \beta X_2\ ,\
\gamma X_1 + \delta X_2
$$
would be reducible to only one, in other words
the derived group would fall into the preceding
case $b)$.
The present composition will therefore be
$$
[X_1, X_2]f = 0\ ,\
[X_1, X_3]f = \alpha X_1 f + \beta X_2 f\ ,\
[X_2, X_3]f = \gamma X_1 f + \delta X_2 f\ ,
\leqno{\rm (A)}
$$
$$
\alpha \delta - \beta \gamma \neq0\ .
$$
We try to see if by changing $X_1 f $ into
$\bar X_1 f = a X_1 f + b X_2 f$, one
can make $\beta=0 $, namely
$[\bar X_1, X_3]f = \rho \bar X_1 f $ ($\rho$ constant).
For this we must have
$$
a (\alpha X_1 f + \beta X_2 f)
+ b (\gamma X_1 f + \delta X_2 f)
= \rho \bar X_1 f = \rho (a X_1 f + b X_2 f)
$$
which reduces to
$$
a (\alpha-\rho) + b\gamma = 0\ ,\
a \beta + b(\gamma-\rho) = 0\ .
\eqno(34)
$$
It therefore suffices for $\rho $ to satisfy
the quadratic equation
$$
\rho^2 -(\alpha+\delta) \rho +\alpha \delta -\beta \gamma =0\ .
\eqno(35)
$$
and from (34) $a,b $ (the ratio $a/b$) can be calculated.
But, from our real point of view, it is necessary to distinguish the
cases where (35) has real or imaginary roots.
Assuming first that they are real, we are reduced to
the composition
$$
[X_1, X_2]f = 0\ ,\
[X_1, X_3]f = \rho X_1 f \ ,\
[X_2, X_3]f = \gamma X_1 f + \delta X_2 f\ ,
$$
where since $\rho\neq0 $, we can make $\rho=1 $ and have
$$
[X_1, X_2]f = 0\ ,\
[X_1, X_3]f = X_1 f \ ,\
[X_2, X_3]f = \gamma X_1 f + \delta X_2 f\ .
\eqno(36)
$$
If $\delta $ which is certainly not zero, is $= 1$,
then when $\gamma\neq0 $, by changing
$X_1 f $ into $1/\gamma\, X_1 f $, we can also make
$\gamma=1 $ and have the composition
$$
[X_1, X_2]f = 0\ ,\
[X_1, X_3]f = X_1 f \ ,\
[X_2, X_3]f = X_1 f + X_2 f\ .
\leqno{\rm Type\ IV:}
$$
and in the case where $\gamma=0 $ the other one
$$
[X_1, X_2]f = 0\ ,\
[X_1, X_3]f = X_1 f \ ,\
[X_2, X_3]f = X_2 f\ .
\leqno{\rm Type\ V:}
$$
When $\delta\neq1 $, by changing $X_2 f $ into
$\bar X_2 f = X_2 f + (\gamma/(\delta-1)\, X_1 f $,
keeping the same
first two equations of the composition (36),
one makes $\gamma=0 $ and has the new type of composition
$$
[X_1, X_2]f = 0\ ,\
[X_1, X_3]f = X_1 f \ ,\
[X_2, X_3]f = h X_2 f\ ,\quad (h\neq 0,1)\ .
\leqno{\rm Type\ VI:}
$$
Finally it remains for us to return to the general equations (A)
to consider the case in which the quadratic
equation (35) has complex roots, so that namely
$$
(\alpha-\delta)^2 + 4 \beta\gamma < 0\ .
\eqno(37)
$$
In this case, if $\alpha $ is not already zero in (A),
we can make it so by
changing $X_1 f $ into
$\bar X_1 f = X_2 f -\gamma/\alpha\, X_1 f $
so that then absorbing the factor $\beta $ into
$X_2 f$,\footnote{
Translator's note:
$$
[\bar X_1, \bar X_3]f
= [X_2, X_3]f - \gamma/\alpha\, [X_1, X_3]f
= \gamma X_1 + \delta X_2 -\gamma/\alpha\, (\alpha X_1+\beta X_2)
=(\delta-\beta\gamma/\alpha) X_2
$$
but $\alpha\delta-\beta\gamma\neq0 $ from (A) so we can make
$\delta-\beta\gamma/\alpha=1$
i.e., ``absorb the factor $\beta $ into $X_2 f$."}
we have
$$
[X_1, X_2]f = 0\ ,\
[X_1, X_3]f = X_2 f \ ,\
[X_2, X_3]f = \gamma X_1 f + \delta X_2 f\ ,
$$
and then by (37)
$$
\delta^2+4\gamma <0\ .
\eqno(38)
$$
Now letting
$$
\bar X_1 f = a X_1 f\ ,\
\bar X_2 f = b X_2 f\ ,\
\bar X_3 f = c X_3 f\ ,
$$
we have
$$
[\bar X_1, \bar X_2]f = 0\ ,\
[\bar X_1, \bar X_3]f = ac/b\, \bar X_2 f \ ,\
[\bar X_2, \bar X_3]f
= bc(\gamma/\alpha\, \bar X_1 f + \delta/b\, \bar X_2 f)\ .
$$
We choose the constants $a,b,c$ in such a way
that one has $ac=b $, $bc\gamma=-a $, which can be done in
a real manner, since by (38) $\gamma $ is negative;
it suffices to take for example:
$a=1, b=c=\sqrt{-1/\gamma}$.
Therefore we have the last type of composition for
the integrable $G_3$'s
$$
[X_1, X_2]f = 0\ ,\
[X_1, X_3]f = X_2 f \ ,\
[X_2, X_3]f = - X_1 f + h X_2 f\ ,
\leqno{\rm Type\ VII:}
$$
$$
{\rm with}\ (h^2<4) \ {\rm by}\ (38)\ ,
$$
(since
$h=c\delta =\delta\sqrt{-\gamma}$,
$h^2/4 = \delta^2/(-4\gamma) <1$).
This type differs from the preceding six because
it does not contain any real invariant one parameter
subgroup, while the others contain at least one.
\section*{\S199. The Two Types of Nonintegrable (simple) $G_3$'s}
For a nonintegrable $G_3$ the derived group
coincides with the $G_3$ so that in the corresponding
equations of composition
\begin{eqnarray}
&&
[X_2, X_3]f = C^1{}_{23} X_1 f + C^2{}_{23} X_2 f + C^3{}_{23} X_3 f
\ ,\nonumber\\ &&
[X_3, X_1]f = C^1{}_{31} X_1 f + C^2{}_{31} X_2 f + C^3{}_{31} X_3 f
\ ,\nonumber\\ &&
[X_1, X_2]f = C^1{}_{12} X_1 f + C^2{}_{12} X_2 f + C^3{}_{12} X_3 f
\ ,
\seteqno{39}
\end{eqnarray}
the determinant
$|C|=\left|\matrix{
C^1{}_{23} & C^2{}_{23} & C^3{}_{23} \cr
C^1{}_{31} & C^2{}_{31} & C^3{}_{31} \cr
C^1{}_{12} & C^2{}_{12} & C^3{}_{12} \cr}\right|
$
{\it is not zero\/}.
Furthermore we can see that the determinant $C$ is
{\it symmetric\/}, namely
$$
C^1{}_{31} = C^2{}_{23}\ ,\
C^1{}_{12} = C^3{}_{23}\ ,\
C^2{}_{12} = C^3{}_{31}\ .
\eqno(40)
$$
This follows from the Jacobi identity
$$
[[X_2,X_3],X_1]f + [[X_3,X_1],X_2]f + [[X_1,X_2],X_3]f = 0\ ,
$$
which by (39) can be written
\begin{eqnarray}
&&
C^2{}_{23} [X_2,X_1] + C^3{}_{23} [X_3,X_1]
+ C^1{}_{31} [X_1,X_2]
\nonumber\\ && \qquad
+ C^3{}_{31} [X_3,X_2]
+ C^1{}_{12} [X_1,X_3] + C^2{}_{12} [X_2,X_3] = 0\ ,
\nonumber
\end{eqnarray}
so that
$$
(C^1{}_{31}-C^2{}_{23}) [X_1,X_2]
+ (C^2{}_{12}-C^3{}_{31}) [X_2,X_3]
+ (C^3{}_{23}-C^1{}_{12}) [X_3,X_1] = 0\ ,
$$
and since the three commutators are linearly
independent, (40) follows from it.
Now we take any two infinitesimal transformations
$X f $, $Y f $ of $G_3$; let
$$
X f = x_1 X_1 f + x_2 X_2 f + x_3 X_3 f\ ,\quad
Y f = y_1 X_1 f + y_2 X_2 f + y_3 X_3 f\ ,
$$
where the coefficients, indicated by
$(x_1,x_2,x_3)$, $(y_1,y_2,y_3) $
will be interpreted as the homogeneous coordinates
of points in a plane,
%\footnote{%*
%Editor's Note: For this and all that follows see
%{\it Analytic Conics\/} by D.M.Y. Sommerville (1924)
%especially p.26 and chapters 11, 12, 13.}
so that every infinitesimal transformation of the
$G_3$ has a point of the plane as an image, from which
it is inversely determined in an unequivocal way.
We examine how the coefficients $(z_i)$ of the commutator
$$
Z f = [X,Y]f
= z_1 X_1 f + z_2 X_2 f + z_3 X_3 f
$$
depend on those $(x_i)$, $(y_i)$ of the factors.
One has
$$
[X,Y]f = (x_2 y_3 - x_3 y_2) [X_2,X_3]f
+ (x_3 y_1 - x_1 y_3) [X_3,X_1]f
+ (x_1 y_2 - x_2 y_1) [X_1,X_2]f
$$
and the three binomials
$$
\xi_1 = x_2 y_3 - x_3 y_2\ ,\
\xi_2 = x_3 y_1 - x_1 y_3\ ,\
\xi_3 = x_1 y_2 - x_2 y_1
$$
are precisely the coordinates of the line which joins the two points
$(x_i)$, $(y_i)$.
From equations (39) one obtains
\begin{eqnarray}
&&
z_1 = C^1{}_{23} \xi_1 + C^1{}_{31} \xi_2 + C^1{}_{12} \xi_3\ ,
\nonumber\\ &&
z_2 = C^2{}_{23} \xi_1 + C^2{}_{31} \xi_2 + C^2{}_{12} \xi_3\ ,
\nonumber\\ &&
z_3 = C^3{}_{23} \xi_1 + C^3{}_{31} \xi_2 + C^3{}_{12} \xi_3\ ,
\seteqno{41}
\end{eqnarray}
and these, since the determinant
$C$ is nonzero and symmetric,
are the equations of an invertible non-degenerate
correlation between the point $(z_i)$ of the plane
and the line $(\xi_i)$.
This correlation is therefore a polarity
with respect to a certain conic $\Gamma$ in the plane,
from which by (41) we immediately write its
equation in the line
coordinates $\xi$
\begin{eqnarray}
&&
C^1{}_{23} \xi_1{}^2
+ C^2{}_{31} \xi_2{}^2
+ C^3{}_{12} \xi_3{}^2
+ (C^1{}_{31} + C^2{}_{23}) \xi_1 \xi_2
\nonumber\\ && \qquad
+ (C^2{}_{12} + C^3{}_{31}) \xi_2 \xi_3
+ (C^3{}_{23} + C^1{}_{12}) \xi_3 \xi_1 = 0
\seteqno{42}
\end{eqnarray}
Therefore:
{\it The image point of the commutator $[X,Y]$ is the
pole, with respect to the fundamental conic $\Gamma$
with the tangential equation (42),
of the line joining the two image points
of the factors $X f$, $Y f$\/}.
Now if the generating transformations $X_1, X_2, X_3$
are replaced by three linearly independent
combinations of themselves, this is equivalent
to performing a transformation of the coordinates
$(x_i)$.
We can take advantage of this to reduce the equation
of the conic $\Gamma$, and consequently
the equations of composition,
to a determined canonical form.
The only distinction to be made is whether or
not the conic $\Gamma$ whose equation has real
coefficients, is real or complex.
\paragraph*{Case l.}
$\Gamma$ is a real conic.
Taking the triangle of reference formed by the two
tangents to the conic and the cord joining the
points of contact,
we will give the equation (42) of $\Gamma$ the canonical form
$ \xi_1 \xi_2 - \xi_2{}^2=0$ so that
$C^3{}_{23} = C^1{}_{12} =1$, $C^2{}_{31}=-2$,
all the other $C$'s are zeros, and we have the
type of composition
$$
[X_1, X_2]f = X_1 f\ ,\
[X_1, X_3]f = 2 X_2 f\ ,\
[X_2, X_3]f = X_3 f\ .
\leqno{\rm Type\ VIII:}
$$
\paragraph*{Case 2.}
If the conic is complex, one takes a triangle
of reference self-conjugate with respect to $\Gamma $
and gives equation (42) the form
$$
\xi_1{}^2 + \xi_2{}^2 + \xi_3{}^2 = 0\ .
$$
Therefore
$C^1{}_{23} = C^2{}_{31} = C^3{}_{12} = 1$
and all the other $C$'s are zero, so
that we have as the last type
$$
[X_1, X_2]f = X_3 f\ ,\
[X_2, X_3]f = X_1 f\ ,\
[X_3, X_1]f = X_2 f\ .
\leqno{\rm Type\ IX:}
$$
The composition VIII is that of the group of motions
of the non-Euclidean plane (hyperbolic geometry),
the other IX that of the group of motions of the
sphere into itself (elliptical geometry) and
they differ in that the type VIII
admits real 2-parameter subgroups, which do
not exist for the type IX (see \S186).
\end{document}
\footnote{
Translator's note:
In the case of the nonintegrable groups, it is much
simpler to note that
$C^{ij}=\frac12 C^i{}_{kl} \epsilon^{jkl}$
are the coefficients of a nonsingular quadratic form.
One can show that nonequivalent sets of structure
constants then correspond to $C^{ij}$'s with different
values of the absolute value of the signature $s$
(absolute value since one can change the sign of
all the $C^{ij}$'s
merely by changing the infinitesimal generators
of the group into their negatives).
Therefore one has two classes of nonequivalent
structure constants: those of
type IX corresponding to
$|s|=0 $
and those of type VIII corresponding to
$|s|=1 $.
One can immediately write down for type IX
$(C^{ij}) = {\rm diag}[1,1,1]$
and for type VIII
$(C^{ij}) = {\rm diag}[1,1,-1]$.
This latter choice for type VIII doesn't
agree with that of Bianchi but is more
useful in practice.}
\end{document}