  We find the exact size of a maximal non-commuting set in unipotent
uppertriangular linear group $UU_4(\mathbb{F}_q)$ in terms of a non-commuting
geometric structure (Refer Definition [10]), where $\mathbb{F}_q$ is the finite
field with $q$ elements. Then we get bounds on the size of such a set by
explicitly finding certain non-commuting sets in the non-commuting structure.
