  This work is motivated by the relation between the KP and BKP integrable
hierarchies, whose $\tau$-functions may be viewed as sections of dual
determinantal and Pfaffian line bundles over infinite dimensional
Grassmannians. In finite dimensions, we show how to relate the Cartan map
which, for a vector space $V$ of dimension $N$, embeds the Grassmannian
${\mathrm {Gr}}^0_V(V+V^*)$ of maximal isotropic subspaces of $V+ V^*$, with
respect to the natural scalar product, into the projectivization of the
exterior space $\Lambda(V)$, and the Pl\"ucker map, which embeds the
Grassmannian ${\mathrm {Gr}}_V(V+ V^*)$ of all $N$-planes in $V+ V^*$ into the
projectivization of $\Lambda^N(V + V^*)$. The Pl\"ucker coordinates on
${\mathrm {Gr}}^0_V(V+V^*)$ are expressed bilinearly in terms of the Cartan
coordinates, which are holomorphic sections of the dual Pfaffian line bundle
${\mathrm {Pf}}^* \rightarrow {\mathrm {Gr}}^0_V(V+V^*, Q)$. In terms of affine
coordinates on the big cell, this is equivalent to an identity of Cauchy-Binet
type, expressing the determinants of square submatrices of a skew symmetric $N
\times N$ matrix as bilinear sums over the Pfaffians of their principal minors.
