Computing isolated orbifolds in weighted flag varieties

Given a weighted flag variety $w\mathrm{\Sigma }(\mu ,u)$ corresponding to chosen fixed parameters $\mu$ and $u$, we present an algorithm to compute lists of all possible projectively Gorenstein $n$-folds, having canonical weight $k$ and isolated orbifold points, appearing as weighted complete intersections in $w\mathrm{\Sigma }(\mu ,u)$ or some projective cone(s) over $w\mathrm{\Sigma }(\mu ,u)$. We apply our algorithm to compute lists of interesting classes of polarized 3-folds with isolated orbifold points in the codimension 8 weighted $G_2$ variety. We also show the existence of some families of log-terminal $\mathbb{Q}$-Fano 3-folds in codimension 8 by explicitly constructing them as quasilinear sections of a weighted $G_2$-variety.