Adjoint representation of a Lie algebra ($\mathrm{ad}$)

The adjoint representation is ${\mathrm{ad}}_x(y)=[x,y]$. For a semisimple Lie algebra, its nonzero weights are the roots and its zero-weight space is the Cartan subalgebra.

The Killing form isIt is symmetric and invariant: $\kappa ([X,Y],Z)=\kappa (X,[Y,Z])$.

The radical ${\mathfrak{g}}^{\perp }=\{x:\kappa (x,\mathfrak{g})=0\}$ of the Killing form is a solvable ideal. Invariance makes it an ideal, and the Cartan solvability criterion applied to its adjoint image proves solvability.

Every invariant bilinear form on a finite-dimensional complex simple Lie algebra is a scalar multiple of its Killing form. A nondegenerate invariant form identifies the algebra with its dual; comparing this identification with the Killing form gives an endomorphism of the irreducible adjoint representation, so Schur lemma makes it scalar.

For ${\mathfrak{sl}}_n(\mathbb{C})$,

Every invariant subspace of the Adjoint representation of a Lie algebra is an ideal. Hence the adjoint representation of a Simple Lie algebra is irreducible.