I was taking class where I encountered that $$Y\_l^mx(-)=\vert j\_1=l,m\_1=m\rangle\vert j\_2=1/2,m\_2=1/2\rangle.$$

I have read [Spin, orbital angular momentum and total angular momentum](https://physics.stackexchange.com/q/184454/) and [Why do we think Spin is angular momentum as opposed to some other quantity?](https://physics.stackexchange.com/q/237486/); however, these did not give me sufficient answers.

My question was that:

1. Why could spin and angular momentum add together? (Quantitatively, they were different as on was caused by spin $\mu$ and the other one was caused by "moving charge". )
2. ACuriousMind mentioned $SO(3)$ group, I suppose what he meant was that $S$ on spin was the same as $L$ on $l,\,m$. However, could you give me some mathematical proof?(I'd like to see how group theory in physics looks like.)
3. (In summary of question 1 and 3) What's the mathematical basis/proof that in the expression $J=L+S$, $L(l,m)$ and $S(j\_2=1/2,s)$ could be numerically equivalently added together in a sense that $$Y\_l^mx(-)=\vert j\_1=l,m\_1=m\rangle\vert j\_2=1/2,m\_2=1/2\rangle?$$