Hi, no unfortunately this is not generally true. This only holds for your specific experiments, where the matrix P_ (cross derivative for x and u) is zero.
In general, it is not true that (- H^-1 * G) == (-G^T * H^-T) , since G = P + B^T * S * A so you are also missing a transpose in above calculations, I think you accidentially dropped it from line 3 to 4.

Ah yes, I did drop the transpose from L^T. (updated OP) While I agree that (- H^-1 * G) == (-G^T * H^-T) is not generally true, isn't (- H^-1 * G)^T == (-G^T * H^-T) just by virtue of the multiplicative transpose property?
In fact we can forego the L^T term altogether in the above analysis and only compare gv and H * lv:
plugging lv = -H^-1 gv directly we have:
gv + H * lv
= gv + H * (-H^-1 gv)
= gv - gv
= 0

Okay, now I see your point. I will need a quiet minute to double check. I will come back to this thread soon!

I quickly looked over the math once again and indeed you are right, those terms cancel to zero!