All Derivations of Semisimple Lie Algebras are Inner
It turns out that all the derivations on a semisimple Lie algebra are inner derivations. That is, they’re all of the form
for some
. We know that the homomorphism
is injective when
is semisimple. Indeed, its kernel is exactly the center
, which we know is trivial. We are asserting that it is also surjective, and thus an isomorphism of Lie algebras.
If we set and
, we can see that
. Indeed, if
is any derivation and
, then we can check that
This makes an ideal, so the Killing form
of
is the restriction of
of the Killing form of
. Then we can define
to be the subspace orthogonal (with respect to
) to
, and the fact that the Killing form is nondegenerate tells us that
, and thus
.
Now, if is an outer derivation — one not in
— we can assume that it is orthogonal to
, since otherwise we just have to use
to project
onto
and subtract off that much to get another outer derivation that is orthogonal. But then we find that
since this bracket is contained in . But the fact that
is injective means that
for all
, and thus
. We conclude that
and that
, and thus that
is onto, as asserted.