Cohomology of solvable saturable pro-$p$ groups and Lie algebras

Let $p$ be an odd prime, and let $n\in \mathbb{N}$ be an integer. We show that the $n^{\text{th}}$ mod-$p$ cohomology of a solvable saturable pro-$p$ group is isomorphic to the $n^{\text{th}}$ mod-$p$ cohomology of its associated ${\mathbb{Z}}_p$-Lie algebra $\text{g}$ as a ${\text{F}}_p$-vector space. Addittonally, we obtain that the $n^{\text{th}}$ mod-$p$ cohomology of $\text{g}$ and of $\text{g}\mathrm{/}p\text{g}$ are isomorphic as ${\text{F}}_p$-vector spaces.