I am reading through the proof of every semisimple Lie Algebra being a direct sum of simple ideals. Firstly, we take an arbitrary ideal $I$ and define $I^{\bot} = $ {$x \in L | \kappa(x,y) = 0 $ for all $ y \in I$}. Now I get why $I \cap I^{\bot} = 0$, but then Humphrey claims that dim $I$ + dim $I^{\bot}$ = dim $L$. I don't see why this is true?
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It's essentially the same argument as for an inner product $x.y = \kappa(x, y)$ on $L$ for ground field $\mathbb{R}$. – anomaly Jun 14 '23 at 16:28
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Also e.g. https://math.stackexchange.com/q/3532873/96384 or https://math.stackexchange.com/q/287187/96384 – Torsten Schoeneberg Jun 14 '23 at 20:29