Proposition 2

For every alternating matrix $R$ of even order $2m$ over a commutative ring $A$, one has

Indeed, since the two members of (8) are polynomials with integer coefficients with respect to the elements of $R$, the principle of prolongation of algebraic identities (chap. IV, § 2, No. 5, Scholium) shows that it is sufficient to give the proof in the case where $A$ is a field of characteristic 0 and where $\det R \neq 0$. If $P$ is an invertible square matrix of order $2m$ over $A$, one has $\det (^tP . R . P) = (\det P)^2 (\det R)$ and $\mathrm{Pf} (^tP . R . P) = (\det P) \mathrm{Pf}(R)$ (prop. 1), so that it is sufficient to prove (8) for $^tP . R . P$ instead of $R$. According to cor. 1 of th. 1, by a suitable choice of $P$, one can suppose that the matrix $^tP . R . P$ is of the form $(\alpha_{ij})$ where all the other $\alpha_{ij}$ being zero (cf. Example). Now the determinant of this matrix is equal to 1, and its Pfaffian also; this completes the proof.