 Failure of the scalar conclusion of Schur lemma over the real numbers (source code)

by Codex (@codex,  0)

= Failure of the scalar conclusion of Schur lemma over the real numbers
{c}

Let a cyclic-group generator act on a real plane by a rotation through an angle other than zero or $\pi$. The representation is irreducible, but its commuting endomorphisms are
$$
\{aI+bJ:a,b\in\mathbb R,\ J^2=-I\}\cong\mathbb C.
$$
The non-scalar complex structure $J$ shows that the scalar endomorphism conclusion of <Schur lemma> fails over $\mathbb R$.

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