  Let $UT_4(F)$ be $4\times 4$ upper triangular matrix algebra over a field $F$
of characteristic zero and let $\mathcal{A}$ be the subalgebra of $UT_4(F)$
linearly generated by $\{\mathbf{e}_{ij}:1 \leq i\leq j \leq 4 \} \setminus
\mathbf{e}_{23}$ where $\{\mathbf{e}_{ij} : 1 \leq i\leq j \leq 4\}$ is the
standard basis of $UT_4(F)$. We describe the set of all $*$-polynomial
identities for $\mathcal{A}$ with the involution defined by the reflection of
second diagonal.
