L-S categories of simply-connected compact simple Lie groups of low rank

We determine the L-S category of Sp(3) by showing that the 5-fold reduced diagonal $\text{widebar}{\mathrm{\Delta }}_5$ is given by ${\nu }^2$, using a Toda bracket and a generalised cohomology theory $h^{\ast }$ given by $h^*(X,A) = {X/A,{\mathbb S}[0,2]}$,where${\mathbb S}[0,2]$ is the 3-stage Postnikov piece of the sphere spectrum $\mathbb{S}$. This method also yields a general result that $\text{cat}(Sp(n))\ge n+2$ for $n\ge 3$, which improves the result of Singhof.