Let $\mathfrak{g}$ be a semisimple Lie Algebra, $\langle \cdot,\cdot\rangle$ the Killing form and $\mathfrak{h}\subset \mathfrak{g}$ a subalgebra of $\mathfrak{g}$. I'm trying to find a counterexample (or prove) the following affirmation: Affirmation: $\mathfrak{h} \cap \mathfrak{h}^\perp = \{0\}$. where $\mathfrak{h}^\perp = \{X \in \mathfrak{g};\ \langle X, Y\rangle =0, \ \forall \ Y\in \mathfrak{h} \}$. Does anyone know how to prove this or have a nice counterexample? NB: In my opinion this affirmation seems false, however all spaces that I tried $\mathfrak{h} \cap \mathfrak{h}^\perp = \{0\}$ holded. Login To add answer/comment