  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.
