= Solution
Periodic integration by parts gives
$$
\int_\Omega(u\mathbin\cdot\nabla v)\mathbin\cdot w\,dx
=
-\int_\Omega(u\mathbin\cdot\nabla w)\mathbin\cdot v\,dx
-\int_\Omega(\nabla\mathbin\cdot u)(v\mathbin\cdot w)\,dx.
$$
Adding one half of the divergence term to both transport forms yields
$$
\boxed{
\langle\widetilde B(u,v),w\rangle
=-\langle\widetilde B(u,w),v\rangle}.
$$
Density extends the identity from smooth fields to all $u,v,w\in\widetilde V$. In particular,
$$
\boxed{\langle\widetilde B(u,v),v\rangle=0}
$$
without requiring $\nabla\mathbin\cdot u=0$.
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