Lie fiber bundles

Question

Let $E$ be a fiber bundle over a manifold $M$ and $\varphi : E \rightarrow TM$ be a morphism of fiber bundles. I call $E$ a Lie fiber bundle if it exists a Lie bracket $[,]$ over the sections of $E$ such that:

1)

$$[s,s']=-[s',s]$$

2)

$$[s,[s',s'']]=[[s,s'],s'']+[s',[s,s'']]$$

3)

$$[f.s,s']=-\varphi (s')(f). s +f.[s,s']$$

with $s,s',s''$ sections of $E$ and $f$ a smooth function over $M$.

Can we make physics calculus over Lie fiber bundles?