This question is an addition to my question on simultaneous diagonalization from yesterday and it is probably also obvious but I just don't know this: Let $G$ be a commutative affine algebraic group over an algebraically closed field $k$. Let $G_s$ be the semisimple part of $G$. Let $rho:G rightarrow GL_n(V)$ be an embedding. Then $rho(G_S)$ is a set of commuting diagonalizable endomorphisms and I know from yesterday that I have unique morphisms of algebraic groups $chi_i: rho(G_s) rightarrow mathbb{G}_m$, $1 leq i leq r$, and a decomposition $V = bigoplus _{i=1}^r E _{chi_i}$, where $E_{chi_i} = lbrace v in V mid fv = chi_i(f)v forall f in rho(G_s) rbrace$. Now, my question is: are the morphisms $chi_i$ independent of $rho$ so that I get well-defined morphisms $chi_i:G_s rightarrow mathbb{G}_m$?
If somebody knows what I'm talking about, then please change the title appropriately! :)
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