All Lie algebras in this question are finite-dimensional and defined over a field $k$ of characteristic $0$ which I'm happy to take to be $\mathbb{R}$ or $\mathbb{C}$.

$\DeclareMathOperator\gr{gr}$Let $L$ be a nilpotent Lie algebra.  It is then filtered by its lower central series, and we have an associated graded nilpotent Lie algebra $\gr L$.  It is definitely not the case that $L$ and $\gr L$ have to be isomorphic; see https://mathoverflow.net/q/122238/174957 for some examples.

Question: what kinds of conditions can I put on $L$ that ensure that it is isomorphic to $\gr L$?  E.g. if the field is $\mathbb{R}$ are the there geometric/topological/algebraic conditions on the associated simply-connected nilpotent Lie group that ensure this?


  [1]: https://mathoverflow.net/q/122238/174957