= Solution
The normalized timelike normal has $n_\alpha n^\alpha=-1$. Multiplying the <linear projections> gives
$$
\begin{aligned}
P^\mu{}_{\alpha}P^\alpha{}_{\delta}
&=(\delta^\mu{}_{\alpha}+n^\mu n_\alpha)
(\delta^\alpha{}_{\delta}+n^\alpha n_\delta)\\
&=\delta^\mu{}_{\delta}+2n^\mu n_\delta
+n^\mu(n_\alpha n^\alpha)n_\delta
=\delta^\mu{}_{\delta}+n^\mu n_\delta.
\end{aligned}
$$
Also $P^\mu{}_{\alpha}n_\mu=n_\alpha+(n_\mu n^\mu)n_\alpha=0$. Thus
$$
\boxed{P^2=P,\qquad P^\mu{}_{\alpha}n_\mu=0.}
$$
The operator fixes every vector tangent to the spacelike hypersurface and annihilates its normal. The repeated-index typo in the TeX is resolved by the actual PDF's matrix-composition indices.
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