On first extensions in $\mathcal{S}$-subcategories of $\mathcal{O}$

We compute the first extension group from a simple object to a proper standard object and, in some cases, the first extension group from a simple object to a standard object in the principal block of an $\mathcal{S}$-subcategory of the BGG category $\mathcal{O}$ associated to a triangular decomposition of a semi-simple finite dimensional complex Lie algebra.