In part (c) it would have been simpler to show that

was non-degenerate simply by inspection, by constructing the inverse (eq. (20)) rather than by calculating the determinant. That was an unnecessary complication, and knowing the value of the determinant doesn't help us in any way here.

Also it's probably worth pointing out explicitly that

is

not a raised version of

, as the notation used throughout much of the book up to this point would suggest. (In fact we can't define raising or lowering of indicies on this manifold, because there's no metric defined (yet)). It's a source of confusion that the same symbol, S, has been used for both, when they

aren't the same tensor. Perhaps Penrose would have been better off using

vs

, or

vs

, or something similar?