Abstract:
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.
Keywords:strong real group, rational group, group of unitriangular matrices.