I think there is an (possible) error in Villani's monograph titled "Hypocoercivity". To be specific, in page 62 (the first snapshot), he defined a new inner product $((\cdot,\cdot))$ as in (8.1). Then in the Theorem 40 below (page 64, with the choice $N = M^3$) this inner product is changed to (8.4), which is definitely different from (8.1). Also, even though we accept (8.4) rather than (8.1) as the definition of $((\cdot,\cdot))$, I don't see why (8.5) implies the advertised conclusion (which basically says the existence of some constant $\lambda$ such that $((h,Lh)) \geq \lambda\,((h,h))$ (please ignore the real part as I am only looking at the real case). (Background information: here $L := A^*A + B$ and $B$ is antisymmetric). I sometimes feel that celebrated mathematicians are so careless and they didn't pay great attention to details. Thanks for any help!