I have a question in the definition of Lie algebra bundle:
In Wikipedia, the definition of Lie Algebra bundle says:
A Lie algebra bundle $\xi = (\xi,p,X)$ is a vector bundle in which each fibre is a Lie algebra and for every $x \in X$, there is an open set $U$ containing $x$, a Lie algebra $L$ and a homeomorphism $$ \phi:U \times L \to p^{-1}(U) $$ such that $\phi_x: x \times L \to p^{-1}(x)$ is a Lie algebra isomorphism.
My Question:
Does the Lie algebra $L$ depend on $x \in X$? i.e., does $L$ changes as $x$ changes, or it remains fixed?
This seems to be a matter of wording to me and I personally would not use a term like "Lie algebra bundle" without making clear what I actually mean. I often use the wording "locally trivial bundle of Lie algebras" to mean that there are local trivializations with values in the product with a fixed Lie algebra. In this case, the individual fibers are indeed all isomorphic to this modelling Lie algebra (at least on a connected manifold).
For the other concept, there is an obvious interpretation as a vector bundle $L\to M$ together with a vector bundle homomorphism $\Phi:\Lambda^2L\to L$ covering the identity map on $M$ such that for each $x\in M$ the value $\Phi_x:L_x\times L_x\to L_x$ satisfies the Jacobi identity. I would rather refer to this a "family of fiber-wise Lie algebra structures" on a vector bundle.