Proposition 1

Let $g$ be a semi-simple Lie algebra and $\varphi$ a finite-dimensional faithful representation of $g$. Then the bilinear form on $g$ associated with $\varphi$ is non-degenerate.

The orthogonal of $g$ with respect to this form is a solvable ideal ($§ 5$, no. 4, Theorem 2) and hence is zero.