  We give minimal presentations for the $RO(C_2)$-graded Bredon cohomology of
the equivariant classifying spaces $B_{C_2}U(n), B_{C_2}SO(n)$ and
$B_{C_2}Sp(n)$ with coefficients in the rational Burnside Green functor
$A_{\mathbf Q}$. This results in an efficient description of rational $C_2$
equivariant Chern, Pontryagin and symplectic characteristic classes. These
classes are then related to each other using the inclusions of maximal tori.
