Example 1

Let $M, N$ be two $g$-modules and $P = \mathscr{L}_K(M, N)$. For an element $f$ of $P$ to be invariant, it is necessary and sufficient, by (6), that $f$ be a homomorphism of the $g$-module $M$ into the $g$-module $N$. In particular, if $M = N$ and $x_M = x_N$ for all $x \in g$, $f$ is invariant if and only if $f$ is permutable with the $x_M$.