For example, for the split form
of
, the 8 level 3-elements and the 28 level 2-elements form an Abelian
subalgebra since there are no elements at levels
(the level is defined in Section 8). This Abelian subalgebra has
dimension 36, which is clearly much greater than the rank (8). We thank Bernard Julia for a discussion on this example. Note
that for subgroups of the unitary group, diagonalizability is automatic.