A characterization of linearly semisimple groups

Let G = SpecA be an affine K-group scheme and A = {w ∈ A*: dim K A*·w < ∞, dim K w· A* < ∞}. Let 〈−,−〉: A* × A → K, 〈w, \( \tilde w \)〉:=tr(w~w), be the trace form. We prove that G is linearly reductive if and only if the trace form is non-degenerate on A*.