Frugal Lazard-divided Lie ring

Definition
A Lazard-divided Lie ring $$L$$ is termed frugal if the following holds: if $$p$$ is a prime number such that $$[[x_1,x_2],\dots,x_p] = 0$$ for all $$x_1,x_2,\dots,x_p \in L$$, then $$t_p(x_1,x_2,\dots,x_p) = 0$$ for all $$x_1,x_2,\dots,x_p \in L$$. In some sense, the Lazard division operations are frugal in terms of departing from giving the zero value.