Finite semisimple Lie algebras and the symmetric adjoint action

A finite-dimensional complex Lie algebra is a finite-dimensional complex vector space $\mathfrak{g}$ with a complex bilinear bracket satisfying $[x,x]=0$ and $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$. A subspace $I$ is an ideal if $[\mathfrak{g},I]\subseteq I$. Put $D^0\mathfrak{g}=\mathfrak{g}$ and $D^{j+1}\mathfrak{g}=[D^j\mathfrak{g},D^j\mathfrak{g}]$, where a bracket of subspaces means the span of all indicated brackets. The algebra is solvable if some $D^j\mathfrak{g}=0$, and semisimple if it has no nonzero solvable ideal. The zero algebra is semisimple. The operator ${\mathrm{ad}}_x(y)=[x,y]$ satisfies $[{\mathrm{ad}}_x,{\mathrm{ad}}_y]={\mathrm{ad}}_{[x,y]}$ by Jacobi. Its Killing form is $B(x,y)={\mathrm{tr}}_{\mathfrak{g}}({\mathrm{ad}}_x{\mathrm{ad}}_y)$; its nondegeneracy in the semisimple case is proved in the next lemma, not assumed here.

The symmetric algebra $S(\mathfrak{g})$ is the free commutative complex algebra generated linearly by $\mathfrak{g}$. Concretely, for a finite vector-space basis $e_1,\dots ,e_n$, put $R_0=\mathbb{C}$, define $R_j=R_{j-1}[e_j]$ for $1\le j\le n$ by the univariate construction of The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, and set $\mathbb{C}[e_1,\dots ,e_n]=R_n$. Its elements are exactly finite complex linear combinations of monomials $e_1^{a_1}\cdots e_n^{a_n}$ with $(a_1,\dots ,a_n)\in {\mathbb{N}}^n$, and multiplication adds exponent tuples. Substituting arbitrary commuting elements for the $e_i$ therefore gives a unique algebra homomorphism, which proves the stated free-algebra property. A change of basis gives inverse linear substitutions, so these descriptions identify canonically by their values on $\mathfrak{g}$. Its grading $S={⨁}_{d\ge 0}{S}_d$ has $S_0=\mathbb{C}$. Each ${\mathrm{ad}}_x$ extends uniquely to a derivation of $S$ by the product rule, and the Lie identities continue to hold because derivations are determined by their values on generators. Define $S^{\mathfrak{g}}=\{p:{\mathrm{ad}}_{x}p=0\text{ for every }x\in \mathfrak{g}\}$ and $S_{+}^{\mathfrak{g}}={⨁}_{d>0}(S^{\mathfrak{g}}\cap S_d)$. The invariant ideal $I=S{S}_{+}^{\mathfrak{g}}$ consists of finite sums of products with positive-degree homogeneous invariants; it is graded and stable under all adjoint derivations. In dimension zero, the iteration is empty, $S=S^{\mathfrak{g}}=\mathbb{C}$ and $I=0$. Only finite basis choices occur in these conventions.