I have been looking at some papers on the "restricted Burnside problem". On page 4 of Vaughan-Lee and Zelmanov's survey, "Bounds in the restricted Burnside problem", I think they implicitly are invoking the claim that:

If $G$ is a Lie algebra over $\Bbb{F}_p$ with nilpotency class $C$, generated by elements $g_1,\dotsc,g_m$, then $\lvert G\rvert \le p^{m^C}$.

How does one prove this claim? Is there a nice combinatorial reason for this (I want to guess that this translates to a nice bound on the dimension of $G$)?