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Let $\mathbb{F}G$ denote the group algebra of a locally finite group $G$ overthe infinite field $\mathbb{F}$ with $char(\mathbb{F})\neq 2$, and let$\circledast:\mathbb{F}G\rightarrow \mathbb{F}G$ denote the involution definedby $\alpha=\Sigma\alpha_{g}g \mapsto\alpha^\circledast=\Sigma\alpha_{g}\sigma(g)g^{\ast}$, where$\sigma:G\rightarrow \{\pm1\}$ is a group homomorphism (called an orientation)and $\ast$ is an involution of the group $G$. In this paper we prove, undersome assumptions, that if the $\circledast$-symmetric units of $\mathbb{F}G$satisfies a group identity then $\mathbb{F}G$ satisfies a polynomial identity,i.e., we give an affirmative answer to a Conjecture of B. Hartley in thissetting. Moreover, in the case when the prime radical $\eta(\mathbb{F}G)$ of$\mathbb{F}G$ is nilpotent we characterize the groups for which the symmetricunits $\mathcal{U}^+(\mathbb{F}G)$ do satisfy a group identity.

Group identities on symmetric units under oriented involutions in group algebras