The previously fastest algorithm for deciding the existence of an independent cut had a runtime of $\mathcal{O}^*(1.4423^n)$, where $n$ is the order of the input graph. We improve this to $\mathcal{O}^*(1.4143^n)$. In fact, we prove a runtime of $\mathcal{O}^*\left( 2^{(\frac{1}{2}-α_Δ)n} \right)$ on graphs of order $n$ and maximum degree at most $Δ$, where $α_Δ=\frac{1}{2+4\lfloor \fracΔ{2} \rfloor}$. Furthermore, we show that the problem is fixed-parameter tractable on graphs of order $n$ and minimum degree at least $βn$ for some $β> \frac{1}{2}$, where $β$ is the parameter.