Let $hat{F}$ be a free pro-$p$ non-abelian group, and let $Delta$ be a commutative Noetherian complete local ring with a maximal ideal $I$ such that $textrm{char}(Delta/I)=p>0$. In [Zu], Zubkov showed that when $p eq2$, the pro-$p$ congruence subgroup $$GL_{2}^{1}(Delta)=ker(GL_{2}(Delta)overset{DeltatoDelta/I}{longrightarrow}GL_{2}(Delta/I))$$ admits a pro-$p$ identity, i.e., there exists an element $1 eq winhat{F}$ that vanishes under any continuous homomorphism $hat{F}to GL_{2}^{1}(Delta)$. In this paper we investigate the case $p=2$. The main result is that when $textrm{char}(Delta)=2$, the pro-$2$ group $GL_{2}^{1}(Delta)$ admits a pro-$2$ identity. This result was obtained by the use of trace identities that originate in PI-theory.
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