I am looking for a reference or an ad-hoc proof of the following fact, which seems to be known to experts: Let $\mathbf{G}$ be a complex algebraic group with maximal (algebraic) torus $\mathbf{T}$ and Weyl group $W$. Let $G=\mathbf{G}(\mathbb{C})$ and let $\mathrm{B}G$ denote the classifying space of $G$; define $T$ and $\mathrm{B}T$ similarly. Then the canonical map between the rational cohomology rings $\mathrm{H}^*(\mathrm{B}G,\mathbb{Q})\to\mathrm{H}^*(\mathrm{B}T,\mathbb{Q})$ is injective and its image is the $W$-fixed elements in $\mathrm{H}^*(\mathrm{B}T)$.

There are numerous proofs of the analogous result for compact (real) Lie groups, i.e. when $G$ is a compact Lie group and $T$ is maximal compact torus in $G$.

It should be possible to deduce the complex case from the real one because every complex reductive Lie group $G$ contains a maximal compact real Lie group $G_0$ which is a retract, but this fact alone is not sufficient as one has to take into consideration the corresponding maximal tori, and make sure that the corresponding Weyl groups of $G$ and $G_0$ relative to these tori are canonically isomorphic.