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

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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 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.

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