= Solution
The projections $P_T$ in the <Gelfand–Tsetlin algebra> satisfy
$$
P_TP_R=\delta_{TR}P_T,\qquad \sum_TP_T=1.
$$
Consequently the map sending a diagonal element to its scalar on each <Gelfand–Tsetlin basis> vector is a <ring isomorphism>
$$
\boxed{GZ_n\cong\prod_T\mathbb C.}
$$
Each factor is a <simple module> over this algebra, and the algebra is their <direct sum> as a module over itself. This proves that $GZ_n$ is a <semisimple algebra>. The argument uses its explicit diagonal structure; being a subalgebra of a <semisimple algebra> by itself would not suffice.
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