 Free step-N nilpotent Lie group

Inside the truncated tensor algebra, let ${\mathfrak{g}}^N(V)$ be the span of iterated Lie brackets of elements of $V$ of lengths at most $N$. Its exponential is the connected simply connected free nilpotent Lie group of step $N$, with tensor multiplication. The bracket is the commutator in the tensor algebra. Its grading defines dilations that multiply degree $k$ by ${\lambda }^k$.