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The two irreducible real modules $U$ and $W$ are nonisomorphic, so the kernel-image argument from part (a), which is valid over any field, givesEvery endomorphism of the one-dimensional trivial module is scalar:

Identify the oriented plane $W$ with $\mathbb{C}$, so that $g$ acts by multiplication by $e^{2\pi i/3}$. A real-linear map commutes with this rotation exactly when it is multiplication by a complex number. Equivalently,where $J$ is rotation through $\pi /2$. Taking $J$ gives a nonscalar intertwining endomorphism of an irreducible real representation. This is the Failure of the scalar conclusion of Schur lemma over the real numbers.