It is conjectured that the Kashiwara-Vergne Lie algebra $\widehat{ \mathfrak{ krv}}_2$ is isomorphic to the direct sum of the Grothendieck-Teichm\"uller Lie algebra $\mathfrak{ grt}_1$ and a one-dimensional Lie algebra. In this paper, we use the graph complex of internally connected graphs to define a nested sequence of Lie subalgebras of $\widehat{ \mathfrak{ krv}}_2$ whose intersection is $\mathfrak{ grt}_1$, thus giving a way to interpolate between these two Lie algebras.
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