# Basis of the Engel algebra

If I have a connected, simply connected nilpotent lie group given by the commutators between the elements of a basis of its Lie algebra how can I recover the left invariant vector fields?

For example I have a Lie group $G$ whose Lie algebra $g$ has dimension 4, with basis $X_1,X_2,X_3,X_4$ and only non zero commutators

$[X_1,X_2]=-X_3$,

$[X_1,X_3]=-X_4$

i.e. the Engel algebra.

An explicit representation of the vector fields in $\mathbb{R}^4$ is $X_1=\partial_1$, $X_2=\partial_2-x_1\partial_3+\frac{x_1^2}{2}\partial_4$ ,$X_3=\partial_3-x_1\partial_4$, $X_4=\partial_4$.

How can i prove this, and what does it mean? I know that for such types of groups the exponential map is diffeomorphism therefore I can identify the lie group and the lie algebra. Then the lie algebra sholud be isomorphic to $\mathbb{R}^4$ as vector spaces, so why do I represent the vector fields as they were on $\mathbb{R}^4$?

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