  In this paper we present an extendible, block gluing $\mathbb Z^3$ shift of
finite type $W^{\text{el}}$ in which the topological entropy equals the
$L$-projectional entropy for a two-dimensional sublattice $L:=\mathbb Z
\vec{e}_1+\mathbb Z\vec{e}_2\subsetneq\mathbb Z^3$, even so $W^{\text{el}}$ is
not a full $\mathbb Z$ extension of $W^{\text{el}}_L$. In particular this
example shows that Theorem 4.1 of [3] does not generalize to $r$-dimensional
sublattices $L$ for $r>1$. Nevertheless we are able to reprove and extend the
result about one-dimensional sublattices for general (non-SFT) $\mathbb Z^d$
shifts under the same mixing assumption as in [3] and by posing a stronger
mixing condition we also obtain the corresponding statement for
higher-dimensional sublattices.
