Being a physicist and only just been introduced to group theory, please forgive my lack of rigour.
I have some confusion regarding Lie groups and Lie algebras, and I want to understand the following example - the addition of angular momentum of two spin $1/2$ particles.
Each Spin-half particle represented by $|\frac{1}{2},\frac{1}{2}\rangle$ and $|\frac{1}{2},-\frac{1}{2}\rangle$. These furnish a representation of the group $SU(2)$. In a sense, the addition of angular momentum, is nothing but $SU(2) \otimes SU(2)$. This is because, in order to add the angular momenta, I must construct a larger hilbert space using a tensor product of the two smaller hilbert spaces of the individual spins.
Now, I've also read that a tensor product of Lie Groups is equivalent to a direct sum of lie Algebras. Hence, $$SU(2)\otimes SU(2) \equiv su(2) \oplus su(2)$$
However the lie algebra is represented by the generators of $SU(2)$, and the above line leads me to believe that if $G_1$ and $G_2$ are the two generators of the two $SU(2)$ groups, then the generator for the combined group should be $G_1\oplus G_2$.
However, from angular momentum algebra, we know that :
$$SU(2)\otimes SU(2) \sim (I_1+G_1)\otimes(I_2+G_2)\sim I_1\otimes I_2 +(I_1\otimes G_2+G_1\otimes I_2)$$
Hence, the new generator of the larger hilbert space, seems to be $$(I_1\otimes G_2+G_1\otimes I_2) \ne G_1\oplus G_2$$
Then what does $su(2) \oplus su(2)$ mean in terms of the generators ? Can someone please explain, where I'm going wrong with the above example ?
Regards.