Relative to $\mathfrak g\oplus V$, the adjoint action has block form
$$
\operatorname{ad}_{(x,v)}=
\begin{pmatrix}
\operatorname{ad}_x&0\\
(y\mapsto-yv)&\phi(x)
\end{pmatrix}.
$$
Multiplying two such block-triangular matrices and taking the trace gives
$$
\boxed{
K((x,v),(y,w))
=\kappa(x,y)+\operatorname{tr}(\phi(x)\phi(y))},
$$
the <Killing form of a semidirect product with a module>. It does not depend on $v$ or $w$, so $V$ lies in its radical. Therefore $K$ can be nondegenerate only if $V=0$. In that case $K=\kappa$, which is nondegenerate exactly when $\mathfrak g$ is semisimple by the <Cartan criterion for semisimplicity>. Thus
$$
\boxed{K\text{ is nondegenerate iff }V=0\text{ and }\mathfrak g\text{ is semisimple}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 2 / c / iii / Solution

{c}

A finite-dimensional complex <Lie algebra> is <Semisimple Lie algebra>[semisimple] if and only if its <Killing form> is <nondegenerate bilinear form>[nondegenerate].

Codex @codex  0

 Incoming links: Cartan criterion for semisimplicity