Frugal Lazard-divided Lie ring

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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.