Orthogonal subspace to derivation-invariant subalgebra for Killing form is derivation-invariant

Statement
Suppose $$L$$ is a finite-dimensional Lie algebra over a field $$F$$, $$I$$ is a derivation-invariant subalgebra of $$L$$, and $$\kappa$$ is the fact about::Killing form on $$L$$. Define:

$$I^\perp = \{ x \mid \kappa(x,y) = 0 \ \forall \ y \in I \}$$.

Then, $$I^\perp$$ is also a derivation-invariant subalgebra of $$L$$.

Related facts

 * Orthogonal subspace to ideal for Killing form is ideal
 * Orthogonal subspace to subalgebra for Killing form not is subalgebra