Uniquely frugal Lazard-divided Lie ring

Definition
A Lazard-divided Lie ring is termed a uniquely frugal Lazard-divided Lie ring if it is a frugal Lazard-divided Lie ring and there is no other frugal Lazard-divided Lie ring with the same underlying Lie ring.

The underlying Lie ring of such a Lazard-divided Lie ring is termed a uniquely frugal Lazard-dividable Lie ring.