Suppose I have the Heisenberg group H say over the $p$-adic integers $\mathbb{Z}_p$, which is the set of $3\times 3$ uni-upper-trianglar matrices over $\mathbb{Z}_p$ . Its Lie algebra $h$ is the set of all $3\times 3$ strictly-upper-trianglar matrices over $\mathbb{Z}_p$.
The commutator relations of the presentation of $H$ carry over exactly to give the Lie bracket for $h$. My question is: does this hold for any arbitrary nilpotent Lie group G in place of the Heisenberg group?