= Solution
All three <vector spaces> now have the same <dimension> $n$. Each of the nine blocks of $P$ commutes with the common <irreducible representation>, so the <Schur lemma> gives
$$
\boxed{P=\begin{pmatrix}
a_{11}I_n&a_{12}I_n&a_{13}I_n\\
a_{21}I_n&a_{22}I_n&a_{23}I_n\\
a_{31}I_n&a_{32}I_n&a_{33}I_n
\end{pmatrix}=A\otimes I_n,\qquad A\in M_3(\mathbb C).}
$$
Conversely $R(g)=I_3\otimes R_1(g)$, so the <Kronecker product> identity immediately verifies commutation for every $A$. The <commutant of a semisimple representation> acts arbitrarily on the three-dimensional multiplicity <vector space> while acting as a scalar on each irreducible factor. Its complex <dimension> is nine.
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