On the Identity Problem for Unitriangular Matrices of Dimension Four

We show that the Identity Problem is decidable in polynomial time for finitely generated sub-semigroups of the group $\mathsf{UT}(4, \mathbb{Z})$ of $4 \times 4$ unitriangular integer matrices. As a byproduct of our proof, we also show the polynomial-time decidability of several subset reachability problems in $\mathsf{UT}(4, \mathbb{Z})$.