  A dimension formula was given in [1] in order to partially classify the Lie
algebras of $S$-unitary type. The natural question of when $\mathfrak{u}_{S}$
and $\mathfrak{u}_{T}$ are isomorphic is left unanswered. In this article, we
will give an answer to this question using the notion of quantum channels and
their Kraus representation. In line with this, we will also discuss linearly
twisted versions of the usual commutator bracket and its relation to the
standard Lie algebra structure on $M_{n}(\mathbb{C})$. Finally, we will mention
some problems that are still unanswered in relation to $S$-unitary type
matrices and twisted brackets.
