Lemma 4

Let $\rho, \sigma, \tau$ be representations of $g$ on vector spaces $M, N, P$. Suppose that we are given a $K$-bilinear mapping $(m, n) \mapsto m.n$ of $M \times N$ into $P$ such that

for all $m \in M, n \in N, x \in g$.

(a) If $m_0 \in M_0$, the mapping $n \mapsto m_0.n$ is a $g$-module homomorphism.

(b) If $n \in N_\delta$, then $m_0.n \in P_\delta$.

(c) If $M$ is a (not necessarily associative) algebra and the $\rho(x)$ are derivations of $M$, $M_0$ is a subalgebra of $M$ and each $M_\delta$ is a right and left $M_0$-module.

For $m_0 \in M_0, n \in N$ and $x \in g$,

whence (a). Assertion (b) follows from (a) (Algebra, Chapter VIII, § 3, no. 4, Proposition 10). If $N = P = M$ and $\sigma = \tau = \rho$, assertion (b) gives assertion (c) as a special case.