Let $M$ be a compact connected semi-simple Lie group. Then by Hopf's Theorem $H^*(M;\mathbb Q)=\Lambda[\omega_1,...,\omega_s]$ where $\omega_i\in H^i(M;\mathbb Q)$ , $i\ge 3$ is odd.

For which $M$, ${\rm dim}(\omega_i)=3$ for all $i=1,...,s$? The examples are $SO(3)$ and quaternionic group $S^3$ (the only simply-connected Lie groups with 1-dimensional torus) as well as their products? Are there any other simply-connected compact Lie groups with this property?

Let the fundamental group $\pi_1 M= (\mathbb Z_2\oplus \cdots\oplus \mathbb Z_2)\oplus (\mathbb Z_{h_1}\oplus \cdots\oplus \mathbb Z_{h_s}) $ be the sum of $s_2$ copies of $\mathbb Z_2$ and $s$ cyclic groups of rank $\ge 3$. Does it imply that ${\rm dim} M\ge 3s_2+5s$?

butis doubly covered by $S^3$." $\endgroup$ – Ben Wieland Jun 24 '14 at 0:28