Given two independent standard normal variables $X, Y\!,$ and $W= X+ Y, Z= X- Y\!,\!$ seek $\mathbb{E}\left [ X+ 2Y\mid Z \right ]$

Question

Given two independent standard normal variables $X, Y\!,$ and $W= X+ Y, Z= X- Y\!,\!$ seek $\mathbb{E}\left [ X+ 2Y\mid Z \right ]\!.$

Attempt. We obtain $$\mathbb{E}\left [ X+ 2Y\mid Z= z \right ]= \mathbb{E}\left [ X+ 2Y\mid X- Y= z \right ]= \mathbb{E}\left [ \left ( Y+ z \right )+ 2Y \right ]= 3\mathbb{E}\left [ X \right ]+ Z= Z\!.$$ Is this correct ?