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In characteristic zero, the connected closed subgroups of $G$ are in 1-1 correspondence with the Lie subalgebras of ${\rm Lie}(G)={\mathfrak g}$, and the Killing form on ${\mathfrak g}$ is non-degenerate. Since the Killing form is $G$-equivariant, it is $H$-equivariant and irreducible summands for non-isomorphic $H$-submodules are orthogonal. Hence the Killing form on ${\mathfrak g}$ restricts to a non-degenerate form on ${\mathfrak g}^H={\rm Lie}(Z_G(H)^\circ)$, so ${\mathfrak g}^H$ is reductive (and hence so is $Z_G(H)^\circ$).