Let $(V,g)$, $(W,h)$ be two vector spaces endowed with symmetric metrics (i.e. symmetric bilinear forms) $g$ and $h$ respectively, possibly degenerated. Let the tensor product $V \otimes W$ be endowed with the tensor product metric $g \otimes h$, that is,$$(g \otimes h)(v \otimes w, v' \otimes w'):= g(v,v')h(w,w').$$ Let $\mathrm{rad}(V)$ be the radical of $g$, that is, the kernel of the map $V \to V^*$, $v \mapsto g(v,-)$, and similarly for $V$ and $V \otimes W$.
I am trying to show that $$ \mathrm{rad}(V \otimes W) = \mathrm{rad}(V) \otimes W + V \otimes \mathrm{rad}(W) .$$ The inclusion $\supseteq$ is obvious but I am having problems to show the opposite $\subseteq$.
Note. I believe the equality should hold for vector spaces but might not for free modules over an arbitrary commutative ring. Any comment on that will also be appreciated.
