  We establish natural splittings for the values of global Mackey functors at
orthogonal, unitary and symplectic groups. In particular, the restriction
homomorphisms between the orthogonal, unitary and symplectic groups of adjacent
dimensions are naturally split epimorphisms. The interest in the splitting
comes from equivariant stable homotopy theory. The equivariant stable homotopy
groups of every global spectrum form a global Mackey functor, so the splittings
imply that certain long exact homotopy group sequences separate into short
exact sequences. For the real and complex global Thom spectra $\mathbf{MO}$ and
$\mathbf{MU}$, the splittings imply the regularity of various Euler classes
related to the tautological representations of $O(n)$ and $U(n)$.
