For example, for the split form E8(8) of E8, the 8 level 3-elements and the 28 level 2-elements form an Abelian subalgebra since there are no elements at levels > 3 (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.