= Solution
Use signature $(-,+,+,+)$, so the future-directed <unit normal> obeys $n^\mu n_\mu=-1$. The <spatial projection tensor> and the <induced metric> are
$$
P^\mu{}_\alpha=\delta^\mu{}_\alpha+n^\mu n_\alpha,
\qquad \gamma_{\alpha\beta}=g_{\alpha\beta}+n_\alpha n_\beta.
$$
Indeed $P^\mu{}_\alpha n^\alpha=0$ and $P^\mu{}_\rho P^\rho{}_\alpha=P^\mu{}_\alpha$: this is an <idempotent linear map> onto the <tangent space> of the spatial <hypersurface>. For a covariant <tensor> both slots must be projected:
$$
\boxed{(\perp T)_{\alpha\beta}=P^\mu{}_\alpha P^\nu{}_\beta T_{\mu\nu}.}
$$
Contracting either slot with $n$ supplies the vanishing factor $P^\mu{}_\alpha n^\alpha$. Thus \b[both normal contractions vanish], without requiring symmetry of $T$.
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