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In adjoint reductive groups $H$ of type $\mathsf{D}$ we show that for everysemisimple element $s$, its centralizer splits over its connected component,i.e., $C_H(s) = C_H(s)^\circ \rtimes \check A$ for some complement $\check A$with strong stability properties. We derive several consequences about theaction of automorphisms on semisimple conjugacy classes. This helps toparametrize characters of the finite groups $\mathsf{D}_{l,\text{sc}}(q)$ and$^2\mathsf{D}_{l,\text{sc}}(q)$ and describe the action of automorphisms onthem. It is also a contribution to the final proof of the McKay conjecture forthe prime 3, see [S21], [S23].

On semisimple classes and component groups in type $\mathsf{D}$