*-polynomial identities of 4x4 upper triangular matrices with the reflection involution

Abstract: 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.