For the classical Lie groups, I think that an easy way to obtain the result is through the fibrations:

SO(n-1)-->SO(n)--> $S^{n-1}$, 

SU(n-1)-->SU(n)--> $S^{2n-1}$

SP(n-1)-->SP(n)--> $S^{4n-1}$

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).