Radical and Derived Group of a Reductive Group
Let \(G\) be a connected reductive group over an algebraically closed field. Then the radical \(R(G)\) is a torus and equals the connected center \(Z(G)^\circ\). The derived group \(G_{\mathrm{der}}=[G,G]\) is semisimple, and multiplication induces an isogeny \[ Z(G)^\circ \times G_{\mathrm{der}} \longrightarrow G. \] Equivalently, \[ G=Z(G)^\circ\cdot G_{\mathrm{der}}, \] with finite central intersection.
Proof
The radical is connected, solvable, and normal. Its unipotent radical would be a connected normal unipotent subgroup of \(G\), so reductivity forces that unipotent radical to be trivial. A connected solvable group with trivial unipotent radical is a torus, and a normal torus in a connected group is central; hence \(R(G)=Z(G)^\circ\).
The quotient \(G/Z(G)^\circ\) has trivial radical, so it is semisimple. The image of \(G_{\mathrm{der}}\) in this quotient is all of it because the quotient has no nontrivial torus quotient. Thus Z(G)^\circ G_{\mathrm{der}}=G. The intersection is a subgroup of the center of the semisimple group \(G_{\mathrm{der}}\), hence finite.