Remark

(1) Theorem 2 admits a converse: if every finite-dimensional representation of g is semi-simple, g is semi-simple. For since the adjoint representation is semi-simple, every ideal of g admits a supplementary ideal and hence can be considered as a quotient of g. If g is not semi-simple then g admits a non-zero commutative quotient and therefore a quotient of dimension 1. Now the Lie algebra K of dimension 1 admits non-semi-simple representations, for example

(2) Let g be a Lie algebra over K and $\sigma$ a representation of g on a vector space M. On the other hand let $f$ be a K-linear mapping of $g$ into M such that:

for all $x, y$ in $g$. By § 1, no. 8, Example 2, being given $\sigma$ and $f$ is equivalent to being given a homomorphism $x \mapsto (f(x), \sigma(x))$ of $g$ into $\mathfrak{af}(M)$. On the other hand we have seen (loc. cit.) that the element $(f(x), \sigma(x))$ of $\mathfrak{af}(M)$ is canonically identified with the element $\rho(x)$ of $\mathfrak{gl}(N)$ (where $N = M \times K$) which induces $\sigma(x)$ on M and maps the element $(0, 1)$ of N to $f(x)$. And $\rho$ is then a representation of $g$ on N such that $\rho(x)(N) \subset M$ for all $x \in g$.

Then, if $g$ is semi-simple, there exists (Lemma 3) a line Z which is supplementary to M in N and annihilated by $\rho(g)$. In other words, there exists an element $m_0 \in M$ such that $(-m_0, 1) \in N$ is annihilated by $\rho(x)$, that is such that

for all $x \in g$.

Suppose that $K = \mathbf{R}$. Let G be a connected Lie group with Lie algebra g. Consider an analytic homomorphism $\phi$ of G into the affine group A of M corresponding to a homomorphism $x \mapsto (f(x), \sigma(x))$ of g into $\mathfrak{af}(M)$. The above results can be interpreted by saying that if g is semi-simple $\phi(G)$ leaves a point of M fixed. For let H be the set of elements of $\mathbf{GL}(N)$ which leave stable all the linear varieties of N parallel to M. There exists (§ 1, no. 8, Example 2) a canonical isomorphism $\psi$ of A onto H. Let Z be a supplementary line of M in N. To say that $\rho(g)$ annihilates Z amounts to saying that $(\psi \circ \phi)(G)$ leaves the points of Z fixed and hence (taking account of the definition of $\psi$) that $\phi(G)$ leaves fixed the projection onto M of the point of intersection of Z and $M \times \{1\}$.