Why must these have integer coefficients?

Question

We are considering diagonal subgroup of classical groups and their lie algebras. We then consider $l=a_1l_1 + a_2l_2 + ...$ where $l_i(H)$, H in the lie algebra, returns the ith entry of H. We then say that if all $a_1$ are integers, $l$ lifts to $e^l(exp(H))=e^{l(H)}$. Why do they need to be integer coefficients?

Answer 1

Hint: There is an easy way to construct matrices $H$ with just two non-zero entries such that $exp(H)=\mathbb I$. For your equation to make sense, you need $e^{l(H)}=1$ for all these matrices.