If , are nonzero then , and will span a subalgebra isomorphic to .
Finding sl(2,C) subalgebras
Statement of theorem
This time, we will prove:
We start by proving a lemma which identifies .
If and then .
Here and is dual to under the Killing form. Recall that:
is the complexification of the Lie algebra of the maximal torus.
is the unique vector such that for all . Uniqueness follows from nondegeneracy of .
Proof of Lemma 1
For a start, let's show that . This is because and , so and we proved last time that .
To show that , we therefore need to prove that for all . We have
Since is a Lie algebra representation, . Therefore
Since for any three matrices , we can cyclically permute to get , so overall:
which is equal to . Since because and . This proves as required.
Interestingly, the only way that depends on and through the scalar factor . This will become important later.
Proof of theorem
Pick and and let .
In fact, by rescaling our choice of (and hence linearly rescaling ), we can assume that . Note that this makes sense because because and is positive definite on (because our group is compact and semisimple).
To check that the subalgebra spanned by , and is isomorphic to , we just need to check that the standard commutation relations hold:
holds by definition.
so and but because was defined by the equation
Therefore . The proof of is similar.
Here, we used the fact that is positive definite on , for which we appealed to the fact that our Lie algebra is the Lie algebra of a compact group. We don't need to do this: it is possible to show that assuming only semisimplicity, but you need to work harder.