Solution
= Solution
The diagonal <Lie algebra> inclusion is $i(a)=(a,\ldots,a)$. Its <dual map> is characterized by
$$
\langle\pi(\eta),a\rangle
=\langle\eta,i(a)\rangle
=a\sum_{j=1}^n\eta_j.
$$
Consequently
$$
\boxed{\pi(\eta_1,\ldots,\eta_n)=\eta_1+\cdots+\eta_n.}
$$