For the classical Lie groups, I think that an easy way to obtain the result is through the fibrations:
$SO(n-1)\to SO(n)\to S^{n-1}$,
$SU(n-1)\to SU(n)\to S^{2n-1}$,
$SP(n-1)\to SP(n)\to S^{4n-1}$
and the homotopy long exact sequence and $\pi_m(S^n) = 0$ for $m$ less than $n$, and $\pi_2(SO(2)) = \pi_2(SU(2))=0$ and the isomorphism of $SP(2)$ and $SO(5)$.
