Let be $E$ a $\mathbb{K}$-vectorial space and $T^{2}E$ the space of $(0,2)$-tensors, $g ^{\otimes 2}: T^{2}E\times T^{2}E\rightarrow \mathbb{K}$ such that $(e_1\otimes e_2,v_1\otimes v_2)\rightarrow g(e_1,v_1)g(e_2,v_2)$ show that if $g\in im(\iota g ^{\otimes 2})$ then $\iota g$ is injective, where $\iota$ is the interior contraction defined like:
$\iota: Bil_{\mathbb{K}}(E,V;\mathbb{K})\rightarrow Lin_{\mathbb{K}}(E,Lin_{\mathbb{K}}(V,\mathbb{K}))$ such that $ f\rightarrow \iota f$ where $\iota f: E\rightarrow Lin_{\mathbb{K}}(V,\mathbb{K})$ $e\rightarrow\iota_e f$ where $\iota_e f:V\rightarrow \mathbb{K}$ $v\rightarrow \iota_e f(v):=f(e,v)$.
In fact this is a if and only if, only that I have already managed to demonstrate a side, something important that I found is that: $ker(\iota g ^{\otimes 2})=${$e_1\otimes e_2 / e_1\in ker(\iota g)\lor e_2\in ker(\iota g)$} then I think it's enough to see that the kernel of $\iota g ^{\otimes 2}$ is {$0$} but that's where I stop.