On a conjecture of Ōshima

The set of homotopy classes of self maps of a compact, connected Lie group G is a group by the pointwise multiplication which we denote by H(G), and it is known to be nilpotent. Ōshima [H. Ōshima, Self homotopy group of the exceptional Lie group G2, J. Math. Kyoto Univ. 40 (1) (2000) 177–184] conjectured: if G is simple, then H(G) is nilpotent of class ⩾rankG. We show this is true for PU(p) which is the first high rank example.