Complemented ideals of ${\mathrm{\ell }}_{\mathrm{\infty }}$

Answering questions raised in \cite{Leonetti, Uzcategui} we characterize ideals $\mathcal{I}\subseteq \mathcal{P}(\omega )$ such that $c_{0,\mathcal{I}}$ is complemented in ${\mathrm{\ell }}_{\mathrm{\infty }}$ as exactly those ideals for which the space $K_{\mathcal{I}}=\mathsf{S}\mathsf{t}\mathsf{o}\mathsf{n}\mathsf{e}(\mathcal{P}(\omega )\mathrm{/}\mathcal{I})$ is approximable, i.e., the unit ball of the space $M(K_{\mathcal{I}})$ of signed Radon measures on $K_{\mathcal{I}}$ is separable in the weak* topology.