For every \(g\in G\), the Jordan decomposition gives \(g=g_sg_u=g_ug_s\). Since \(G\) is commutative, the semisimple and unipotent parts of products are products of the corresponding parts: \[ (gh)_s=g_sh_s,\qquad (gh)_u=g_uh_u. \] Thus \(G_s\) and \(G_u\) are subgroups and every element has a unique product decomposition with factors in them.

Closedness follows by embedding \(G\) into a diagonal-plus-unipotent normal form inside some \(\operatorname{GL}(V)\): semisimple and unipotent parts are obtained by polynomial functions on the finite algebra generated by the elements. The product map is bijective with inverse \(g\mapsto(g_s,g_u)\), and these maps are morphisms. The semisimple commutative subgroup is simultaneously diagonalizable, while \(G_u\) is unipotent by definition.