I'm pretty sure that the following (if true) is a standard result in linear algebra but unfortunately I could not find it anywhere and even worse I'm too dumb to prove it: Let $k$ be a field, let $V$ be a finite-dimensional $k$-vector space and let $S \subseteq \mathrm{End}\_k(V)$ be a subset of pairwise commuting (i.e. $\lbrack S, S \rbrack = 0$) endomorphisms. Then the following holds:


1. If all $f \in S$ are diagonalizable, then there exist maps $\chi\_i:S \rightarrow k$, $i=1,\ldots,r$, such that $V = \bigoplus\_{i=1}^r E\_{\chi\_i}(S)$, where $E\_\chi(S) := \lbrace v \in V \mid fv = \chi(f)v \ \forall \ f \in S \rbrace$.
2. The maps $\chi\_i$ in 1 are unique.
3. 1 is equivalent to the existence of a basis $\mathcal{B}$ of $V$ such that for each $f \in S$ the matrix $M\_{\mathcal{B}}(f)$ of $f$ with respect to $\mathcal{B}$ is diagonal. (I believe that this might not be true)
4. If all $f \in S$ are trigonalizable, then there exists a basis $\mathcal{B}$ of $V$ such that for each $f \in S$ the matrix $M\_{\mathcal{B}}(f)$ of $f$ with respect to $\mathcal{B}$ is upper triangular and for each diagonalizable $f \in S$ the matrix $M\_{\mathcal{B}}(f)$ is diagonal.


I know that a set of commuting diagonalizable endomorphisms can be simultaneously diagonalized in the sense of 3 but I don't know how to prove 1 (my problem is the "glueing" of the $\chi$-maps when I try to prove this by induction on $\mathrm{dim}V$). Also, I know that the first part of 4, the simultaneous trigonalization, holds but I don't know how to show that there exists a basis which then also diagonalizes all diagonalizable endomorphisms. This should follow from 1, I think.


Perhaps, because all this is probably standard stuff, I should mention that this is not a homework problem :)


One additional question: Suppose that $k$ is algebraically closed and that $G$ is an affine commutative algebraic group over $k$ which coincides with its semisimple part, embedded as a closed subgroup in some $GL(V)$. Are the maps $\chi\_i:G \rightarrow \mathbb{G}\_{m}$ morphisms of algebraic groups?



Simultaneous diagonalization
All of these are true. First note that the space of endomorphisms of $V$ is finite-dimensional, so even an infinite $S$ can just be replaced by finitely many matrices that have the same span (it's really more elegant to think about the span of $S$ as a Lie algebra, rather than $S$ itself). You actually may want to look at some discussion of abelian Lie algebras, since really your question is about the natural structure theorem for semi-simple representations of abelian Lie algebras (if you think in this language question 4) is obvious from the first 3, since any representation has a flag whose successive quotients are semi-simple).


The important point for proving 1) is that if A and B commute and are both diagonalizable, you should analyze the action of B on the eigenspaces of A. The spaces $E\_\chi$ above are the eigenspaces of B's action on each eigenspace of A (and if there were a third matrix, you would take the eigenspaces of C acting on the eigenspaces of B in the eigenspaces of A, etc.). 2) is clear, and for 3) just pick any basis of these iterated eigenspaces.


For 3) => 1), you have associated to each basis vector a $\chi$, given by looking at how the elements of $S$ act on it. $E\_\chi$ is just the span of all vectors associated to the particular map $\chi$.


For 4), there is a similar argument, but you have to use a flag (the $i$-th subspace being things killed by $(A-\lambda I)^i$ for some scalar $\lambda$) rather than a subspace decomposition. Still, matrices commuting means that B will preserve this flag, so just as one refined the eigenspaces, one can refine this flag.


