Level-two Virasoro descendant Gram matrix ($G_2=\left(\begin{array}{cc}4h+c/2& 6h\\ 6h& 4h(2h+1)\end{array}\right)$)

For a normalized highest-weight vector with $L_0|h⟩=h|h⟩$, $L_{n>0}|h⟩=0$ and $L_n^{†}=L_{-n}$, the level-two descendants $L_{-2}|h⟩,L_{-1}^2|h⟩$ have the displayed Gram matrix. It follows from the Virasoro algebra: $⟨L_2{L}_{-2}⟩=4h+c/2$, $⟨L_2{L}_{-1}^2⟩=6h$ and $⟨L_1^2{L}_{-1}^2⟩=4h(2h+1)$. At $h=-1,c=26$ its null direction is $(1,3/2)$, giving the level-two scalar Virasoro null state. Nullity of a descendant is a metric statement; satisfying all physical constraints must be checked separately.