Hey I'm stuck on the last portion of a question

Right I've done it up to the point of showing thatQuote:

For a plane gravity wave of the form

Show that the linearised gravitational field equations require that

Where

and

Hence show that for the case that is an eigenvector of the Riemann tensor in the sense that

So I've reduced it to showing that which I can't really seem to get out, any hints would be much appreciated

Thanks!