  We give the construction of weighted Lagranngiann Grassmannians$wLGr(3,6)$
and weighted partial $A_3$ flag variety $wFl_{1,3}$ coming from the symplectic
Lie group $Sp(6,\mathbb C)$ and the general linear group $GL(4,\mathbb C)$
respectively. We give general formulas for their Hilbert series in terms of Lie
theoretic data. We use them as key varieties (Format) to construct some
families of polarized 3-folds in codimension 7 and 9. At the end, we list all
the distinct weighted flag varieties in codimension $4\le c\le 10$.
