A Lie algebra is an algebra with a Lie bracket. I have example for Lie algebras For example if $V$ is a vector space over a Field $\mathbb{F}$ then $\mathcal{L}(V)$ forms a Lie algebra. We have definition of locally trivial smooth Lie algebra bundles similar to that of a vector bundle
Let $xi, M$ be smooth manifolds and $\pi : \xi \rightarrow M$ be a smooth surjective map satisfying :
For every $m$ in $M,~ \xi_{m}=\pi^{-1}(m)$ is a Lie algebra.
For each $m$ in $M$ there exist an open subset $U$ of $M$ containing $m$, a Lie algebra $L$ and a diffeomorphism $\phi : \pi^{-1}(U) \rightarrow U \times L$ such that for each $n \in U$ the restriction of $\phi$ to $\xi_{n}$ is isomorphic to $L$ as a Lie algebra.
I want to construct some concrete examples for this. I was trying to think of some structure whose each fibre is $\mathcal{L}(V)$. But I am not getting the picture. I also want to know where Lie algebra a bundles are used. I am not getting books regarding these concepts.
Apart from, of course, trivial examples the reason why you do not find much material about this is connected to terminology.
Lie algebra bundles can be identified with totally intransitive Lie algebroids, and it is under this form that you're more likely to see them mentioned, for example in the book by
K.C.H. Mackenzie: General theory of Lie groupoids and Lie algebroids, Cambridge University Press 2005.
There you may find various examples: each time you have a transitive Lie algebroid the kernel of the anchor defines a Lie algebra bundle on the base.