Quantization of angular momentum in $SO(3)$

Question

When hermitian operators $L_1, L_2, L_3$ follow the commutation relations:

$$ [L_1,L_2]=i\;L_3 \\ [L_2,L_3]=i\;L_1 \\ [L_3,L_1]=i\;L_2 $$

one can show that, assuming they are in finite number, their eigenvalues are integer or half-integer. It turns out that the basis of the Lie algebra of $SU(2)$ follows these relations, and after identifying the angular momentum to these operators, it constitutes a proof of the quantization of angular momentum.

Now, my mathematical knowledge does not go much further, but the basis of the Lie algebra of $SO(3)$ follows the same commutation relations, with the exception of the $i$ coefficient. In that case, I cannot find any way to obtain quantized eigenvalues. How come I read here and there that $SO(3)$ and $SU(2)$ have isomorphic Lie algebras, and that they basically lead to the same quantization? If so, how do I derive that quantization using only $SO(3)$ operators? And if the result is the same, why would we bother using $SU(2)$ when we can use real 3x3 rotation operators?

Answer 1

su(2) and so(3) Lie algebra are homomorpic, so if you redefine L by a factor "-$i$" then you get the so(3). But the group SU(2) is cover group to SO(3). its odd dimension representation matrices correspondences to all SO(3) representation matrices, which means SO(3) has no even dimension representations. So for paricles with integer angular momentun of course you can use SO(3), but for half integer spins, Like spin-1/2 fermions you have to use SU(2) representation. Altogether, using SU(2) is valid for all cases while SO(3) not. So this may provide you some idea?