Let $G$ be a split reductive group over a field of characteristic zero. Is there always an isogeny $G \to H \times T$ with $H$ semi-simple and $T$ a split torus ?

I have in mind the case of $\text{GL}_n$ where we have (if I am not mistaken) the isogeny : $$\text{GL}_n \to \text{PGL}_n \times \mathbf{G}_m,\;\; g \mapsto (\overline{g},\det(g)).$$

Maybe it's even true that there is always a morphism $G \to G^{ad} \times Z(G)$ but I only need the less precise result I asked above.

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    $\begingroup$ Are you assuming that $G$ is connected? There is always a canonical isogeny to $G^{ad}\times G^{ab}$, where $G^{ab} = G/G^{der}$. $\endgroup$ – Keerthi Madapusi Pera Jan 11 at 19:27
  • $\begingroup$ I am indeed assuming that $G$ is connected and your comment answers my question. Thanks a lot ! $\endgroup$ – user134482 Jan 12 at 17:31
  • $\begingroup$ Note that the structure of a connected ereductive group may be rather complicated, when $G$ isn't semisimple or a torus: for examole, $D_4$ has a center of order 4 but exponent 2 and therefore can be combined with a torus in a complicated way. Even so, an isogeny of the sort mentioned by Pera is always possible. $\endgroup$ – Jim Humphreys Jan 12 at 17:33

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