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).

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 $SO(5)$.

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,

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).

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).