Given an exceptional Lie group $G$ and maximal torus $T$ thereof, the inclusion $T \hookrightarrow G$ induces a map $BT \to BG$ of classifying spaces and a cohomological pullback $$H^*(BG) \cong H^*(BT)^W \hookrightarrow H^*(BT)$$ as the subring of Weyl group invariants. Except for that of $G_2$, these Weyl groups range from "large" to absurdly large, but it seems vaguely possible that these inclusions of polynomial rings might nevertheless be comprehensible. Is that the case?
Or if, more likely, explicit generators for $H^*(BG)$ wouldn't fit in this answer space, where could I find a reference for these inclusions?
NB: I am completely happy taking coefficients in $\mathbb Q$.