Derivation of Equation 26 in Bardeen's Four Laws of Black Hole Thermodynamics

[thatboi]

Hi all,
I am currently reading Bardeen's Paper on The Four Laws of Black Hole Thermodynamics: https://projecteuclid.org/journals/...ws-of-black-hole-mechanics/cmp/1103858973.pdf
and am struggling with the derivation of equation 26. Specifically, I do not see how he uses the Lie derivative statement just above the equation to make the 2 terms disappears in the second equality because one of the terms involves the differential of n as opposed to l.

 

[ergospherical]

Hm, that's indeed quite a headache. The little comment about the Lie derivative looks like it contains a typo (!), ought to be $(L_{l} \delta l)^a = l^b \nabla_b (\delta l)^a - (\delta l)^b \nabla_b l^a = 0$, no?