  We study a presentation of Khovanov - Lauda - Rouquier's candidate
$2$-categorification of a quantum group using algebraic rewriting methods. We
use a computational approach based on rewriting modulo the isotopy axioms of
its pivotal structure to compute a family of linear bases for all the vector
spaces of $2$-cells in this $2$-category. We show that these bases correspond
to Khovanov and Lauda's conjectured generating sets, proving the non-degeneracy
of their diagrammatic calculus. This implies that this $2$-category is a
categorification of Lusztig's idempotent and integral quantum group
$\bf{U}_{q}(\mathfrak{g})$ associated to a symmetrizable simply-laced Kac-Moody
algebra $\mathfrak{g}$.
