How represent angular momentum in representation $(a,b)\otimes (c,d)=(a\otimes c,b\otimes d)$ ?
I also post Tensor product of representations of Lorentz group
Agreements: $I_a$ is $(2a+1)$-dimensional matrix, and $J_b=J^{(b)} $ (index $i$ is removed)
I try
$$(a,b)=I_a\otimes J_b +J_a\otimes I_b,$$ by definition.
$\pi_1\otimes \pi_2(X)=\pi_1(X)\otimes I_{dim(\pi_2)}+I_{dim(\pi_1)}\otimes \pi_2(X)$
then
$$(a,b)\otimes(c,d)(J)=J_{(a,b)}\otimes I_{(a,b)}+I_{(a,b)}\otimes J_{(c,d)}=$$ $$=(I_a\otimes J_b +J_a\otimes I_b)\otimes I_c\otimes I_d+I_a\otimes I_b\otimes (I_c\otimes J_d +J_c\otimes I_d)=$$ $$=J_a\otimes I_b\otimes I_c\otimes I_d+I_a\otimes J_b\otimes I_c\otimes I_d+I_a\otimes I_b\otimes J_c\otimes I_d+I_a\otimes I_b\otimes I_c\otimes J_d$$
but,
$$(a\otimes b,c\otimes b)=I_{a\cdot b}\otimes J_{c\cdot b}+...$$
where $I_{a\cdot b}$ is unit matrix with dimension $(2a+1)(2b+1)$, and $J_{c\cdot b}$ is representation of angular momentum operator in $(2c+1+2d+1)$ dimension.
And we have $J_{cd}\neq I_c\otimes J_d$
How ?
