Definition 1

Let $g$ be a Lie algebra, $g'$ a subalgebra of $g$ and $\rho'$ a representation of $g'$ on $V'$. A representation $\rho$ of $g$ on $V$ is called an extension of $\rho'$ to $g$ if there exists an injective homomorphism of the $g'$-module into the $g'$-module $V$. We also say that the $g$-module $V$ is an extension of the $g'$-module $V'$.

If $\rho'$ is finite-dimensional and $g'$ is a solvable ideal of $g$, it is necessary for the existence of a finite-dimensional extension that $[g, g']$ be contained in the largest nilpotency ideal of $\rho'$ (§ 5, no. 3, Theorem 1).