= Commutant of a semisimple representation
{title2=$\operatorname{End}_G(V)=\bigoplus_\lambda M_{m_\lambda}(\mathbb C)$}
For a finite-dimensional complex <semisimple representation> $V=\bigoplus_\lambda(\mathbb C^{m_\lambda}\otimes V_\lambda)$ with pairwise inequivalent <irreducible representations> $V_\lambda$, its commuting <endomorphisms> are exactly $\bigoplus_\lambda(A_\lambda\otimes I_{V_\lambda})$. To prove it, write each block as an <intertwiner> and apply <Schur's lemma>: blocks between inequivalent constituents vanish, and blocks between fixed copies of the same constituent are scalar. The dimension is $\sum_\lambda m_\lambda^2$.
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