Translation length is conjugacy invariant and homogeneous on powers

If $g$ is elliptic, [L1] gives a fixed vertex, so every power $g^n$ fixes that same vertex and has translation length $0$. Conjugating by $h$ carries a fixed vertex of $g$ to a fixed vertex of $hg{h}^{-1}$, so conjugation preserves the elliptic case and its translation length.

If $g$ is hyperbolic with axis $A_g$ and translation length $\mathrm{\ell }(g)$, then $hg{h}^{-1}$ preserves the line $h{A}_g$ and translates it by the same distance, so $\mathrm{\ell }(hg{h}^{-1})=\mathrm{\ell }(g)$. Also $g^n$ preserves $A_g$ and shifts every vertex on it by $\mathrm{\mid }n\mathrm{\mid }\mathrm{\ell }(g)$, so $\mathrm{\ell }(g^n)=\mathrm{\mid }n\mathrm{\mid }\mathrm{\ell }(g)$ for $n\mathrm{\ne }0$.

The case $n=0$ gives $\mathrm{\ell }(1)=0$, already covered by step 1.1, so the displayed formulas hold for every integer $n$.