Computing commutators in Lie algebras

[Silviu]

Hello! Is there any rule to do sums and products like the one in the attached picture (Lie.png) without going through all the math theory behind? I understand the first (product) and last (sum) terms, but I am not sure I understand how you go from one to another.
Thank you!

 

[Orodruin]

You can split product representations into irreps by using Young tableaux, but I do suggest you learn the theory behind it and why it works.

 

[ChrisVer]

especially with SU(2)'s you can just add spins...
2= spin 1/2
3= spin 1
etc
and then seeing the multiplicity of the final spins

Eg your example has 3 spin 1/2 particles, so you'd get something like:
(1/2 1/2) 1/2 = ( 1 0 ) 1/2 = (1 1/2) (0 1/2) = 3/2 1/2 1/2
so a 4 2 2