Pfaffian ($\mathrm{Pf}(A{)}^2=\det A$)

For an antisymmetric $2n\times 2n$ matrix, the Pfaffian is $\mathrm{Pf}(A)=(2^{n}n!{)}^{-1}{\sum }_{\sigma \in S_{2n}}\mathrm{sgn}(\sigma ){\prod }_{j=1}^n{A}_{\sigma (2j-1),\sigma (2j)}$. Equivalently, it is the coefficient of the top exterior power of the associated alternating form, with the usual factorial normalization. Its square is the determinant. In size four it is $a_{12}{a}_{34}-a_{13}{a}_{24}+a_{14}{a}_{23}$, so its vanishing characterizes the simple nonzero bivectors of the Klein quadric.