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In this paper, we prove a generalization of a discreteness criteria for alarge class of subgroups of PSL$_2(\mathbb{C})$. In particular, we show thatfor a given finitely generated, purely loxodromic, free Kleinian group$\Gamma=\langle\xi_1,\xi_2,\dots,\xi_n\rangle$ for $n\geq 2$, the inequality  $$\left|\text{trace}^2(\xi_i)-4\right|+\left|\text{trace}(\xi_i\xi_j\xi_i^{-1}\xi_j^{-1})-2\right|\geq2\sinh^2\left(\frac{1}{4}\log\alpha_n\right)$$  holds for some $\xi_i$ and $\xi_j$ for $i\neq j$ in $\Gamma$ provided thatcertain conditions on the hyperbolic displacements given by $\xi_i$, $\xi_j$and their length $3$ conjugates formed by the generators are satisfied. Above,the constant $\alpha_n$ turns out to be the real root strictly larger than$(2n-1)^2$ of a fourth degree, integer coefficient polynomial obtained bysolving a family of optimization problems via Karush-Kuhn-Tucker theory. Theuse of this theory in the context of hyperbolic geometry is another novelty ofthis work.

Isometries of length $1$ in purely loxodromic free Kleinian groups and trace inequalities