# How to show that two operators in the forms of vectors commute?

This is an exercise from Peskin&Schroeder's book. The exercise requires to show that $\textbf{J+}$ and $\textbf{J-}$ commute with each other. What is the exact meaning of commutation between operators that are the form of vectors? Do I have to just show that $[\textbf{J+}^i, \textbf{J-}^i]=0$? Or do I have to show that $[\textbf{J+}^i, \textbf{J-}^j]=0$? I am confused...

• Are you confusing 'vector operator' (i.e. an operator that acts on vectors) with 'operator that is a vector'? – Acccumulation Mar 15 '18 at 15:17
• I mean the latter. – Keith Mar 15 '18 at 15:18

in this specific case you have to show that $$[J_+^i,J_-^j]=0$$ meaning that every component commutes with every component.
• The above relation is more general than $[J_+^i,J_-^i]=0$. – G K Mar 15 '18 at 15:15