Parity operators and anti commutators

[qtm912]

I am trying to understand the following which is proving difficult:

It is found that (and the proof here is clear)

[P, Jj] anticommutes with Vi

Where P = parity operator
Jj and Vi are the j th and i th components of the angular momentum vector and an arbitrary vector respectively.

It is then stated that because P has the same anticommuting property, that [P,Jj] must be proportional to P ,ie that

[P,Jj] = λP

Where λ is a scalar and the same for all j

I am unclear how this is imputed. Why should they be proportional.

Thanks in advance

(ref is Binney and Skinner QM book Chapter 4 page 66

 

[azntoon]

)This statement is simply a consequence of the fact that the parity operator anticommutes with any vector. Since [P,Jj] also anticommutes with Vi, it follows that it must also anticommute with P. This means that the two operators must be proportional to each other, i.e. [P,Jj] = λP. The scalar λ is the same for all j because the angular momentum components are all related by the same symmetry transformation (rotations).