I think the vector approach would be more simple to understand .

Suppose , wrt any origin chosen , the P.V. of the first and second particles be r1 and r2 .

Given that r1-r2= d  + ap  .............( 1) ( a is a scalar which depends on time ) ( p and -p their respective momentum )

Now denote the angular momentum of any particle by L

So , L1= r1 X p

L2= r2 X( -p) )

So L1+ L2 = ( r1-r2) X p = ( d+ a p ) Xp = d Xp = a constant ( independent of time )