(XiXj) = Σk=rk=1 cijk Xk,
where the c’s are constants. Moreover, from Jacobi’s identity and the identity (XY) + (YX) = 0 it follows that the c’s are subject to the relations
and
cijt + cjit = 0,
Σs (cjks cist + ckis cjst + cijs ckst) = 0
(iii.)
for all values of i, j, k and t.
The fundamental theorem of the theory of finite continuous groups is now that these conditions, which are necessary in order Determination of the distinct types of continuous groups of a given order. that X1, X2, ..., Xr may generate, as infinitesimal operations, a continuous group of order r, are also sufficient.
For the proof of this fundamental theorem see Lie’s works (cf. Lie-Engel, i. chap. 9; iii. chap. 25).
If two continuous groups of order r are such that, for each, a set of linearly independent infinitesimal operations X1, X2, ..., Xr and Y1, Y2, ..., Yr can be chosen, so that in the relations
(XiXj) = Σcijs Xs, (YiYj) = Σ dijs Ys,