  We compute the Chow-Witt rings of the classifying spaces for the symplectic
and special linear groups. In the structural description we give, contributions
from real and complex realization are clearly visible. In particular, the
computation of cohomology with $\mathbf{I}^j$-coefficients is done closely
along the lines of Brown's computation of integral cohomology for special
orthogonal groups. The computations for the symplectic groups show that
Chow-Witt groups are a symplectically oriented ring cohomology theory. Using
our computations for special linear groups, we also discuss the question when
an oriented vector bundle of odd rank splits off a trivial summand.
