I’m keeping the same notation as all the previous posts here on semisimple Lie algebras.

Consider the real vector space

I claim that the form (obtained by the isomorphism induced by the Killing form and the Killing form itself) is actually an inner product making into a euclidean space. To see this, we will check that for all . Indeed:

where is the Killing form, by definition.

Now

Now acts by the scalar on , so after dividing by , this becomes

But as we showed yesterday, , so the sum in question is actually positive. This proves one half of:

Proposition 1is a euclidean space and . (more…)