Question

Prove that if an operator commutes with any two components of J it
must also commute with the third component.

Answer 1

We know, that [Jx,Jy]=ihJz
So, [Jz,H]=[[Jx,Jy],H]=[JxJy-JyJx, H]=Jx[Jy,H]+[Jx,H]Jy-Jy[Jx,H]-[Jy,H]Jx=0, because of the "an operator commutes with any two components of J",

We prove that [Jz,H]=0, H - operator