$\mathcal L$ and $\mathcal L'$ be full rank lattices in $\mathbb R^n$ with shortest vectors $v_1,\dots,v_n$ and $v_1',\dots,v_n'$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$ $$\|v_1'\|_2\leq\dots\leq\|v_n'\|_2$$ hold and $\mathcal L'=Span(v_1,\dots,v_n)$ and $\mathcal L=Span(v_1',\dots,v_n')$.

Do the shortest vectors of $\mathcal L\otimes\mathcal L'$ have any relation to shortest vectors of $\mathcal L$ and $\mathcal L'$ and what is the span of $v_i\otimes v_j'$?

Do maximal lattices in $\mathcal L\otimes\mathcal L'$ have any relation to $\mathcal L$ and $\mathcal L'$ and $v_1,\dots,v_n$ and $v_1',\dots,v_n'$?