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

L-S Categories of Simply-connected Compact Simple Lie Groups of Low Rank