Normal elements of completed group algebras over ${\mathrm{S}\mathrm{L}}_3({\mathbb{Z}}_p)$

Let $p$ be a prime integer and ${\mathbb{Z}}_p$ be the ring of $p$-adic integers. By a purely computational approach we prove that each nonzero normal element of a completed group algebra over the special linear group ${\rm SL}_3($\mathbb{Z}$_p)$ is a unit. This give a positive answer to an open question in \cite{WeiBian2} and make up for an earlier mistake in \cite{WeiBian1} simultaneously.