The strong reality and rationality of groups of unitriangular matrices of order $\le 8$ over fields of characteristic 2

It is proved that an arbitrary matrix from the group of unitriangular matrices $UT_n(K)$, $n\le 8$, over an arbitrary field $K$ is conjugate in this group to a matrix whose commutativity graph is a forest. From this fact we derive the strong reality and rationality of the group $UT_n(K)$ for $n\le 8$ over an arbitrary field $K$ of characteristic 2.