I'm following a book in Humphrey's Introdutction to Lie Algebra's and Representation Theory. I'm reading the proof that a semisimple Lie algebra is the direct sum of simple modules.
It uses the following observation without proof: If $L$ is a semisimple Lie algebra and $I \subset L$ is a subspace (it's actually an ideal here), then $\dim I + \dim I^\perp = \dim L$.
Why is that true?
This is ofcourse familiar if the Killing from is replaced by an inner product.
What exactly is needed from a bilinear form for this to be true? (Nondegenericity I guess, being symmetric too maybe)
Actually, in this proof it's enough to show that $\dim I + \dim I^\perp \geq \dim L$ so
If there's an easy proof of the $\geq$ direction it would be interseting too.