I'm specifically wondering about the equation
$\Box R_{abcd} = 0$
If this equation holds in some manifold, is it also true that
$\Box {R^a}_{bcd} = 0$ ?
It's not obvious to me, because contraction with the metric doesn't appear to commute with the differential operator. Yet conceptually, it seems like all we're doing is replacing a vector with its co-vector, and if the vector has zero Laplacian then surely the co-vector does as well?