Plücker embedding ($W=\mathrm{span}(u_1,\dots ,u_k)\mapsto [u_1\wedge \cdots \wedge u_k]$)

The Plücker embedding maps a $k$-plane to the projective class of the exterior product of a basis. A change of basis multiplies this product by its determinant, so the projective class is well defined. The plane is recovered as the vectors whose exterior product with this element is zero. For two-planes in ${\mathbb{C}}^4$, its image in ${\mathbb{CP}}^5$ is the Klein quadric.

A line in real projective space ${\mathbb{P}}^3$ is the span of two independent homogeneous vectors $A,B$. Its Plücker coordinates are the entries of the displayed rank-two antisymmetric matrix, determined up to a nonzero scalar. They satisfy $L^{01}{L}^{23}-L^{02}{L}^{13}+L^{03}{L}^{12}=0$. With a chosen volume orientation, its dual plane form is $L_{ij}=\frac{1}{2}{\epsilon }_{ijkl}{L}^{kl}$. This is projective duality using the volume form, not lowering indices with a Euclidean metric.

The intersection of a line in Plücker coordinates with a plane covector $F$ is the first displayed homogeneous vector: it lies in the span of the defining points and its contraction with $F$ is zero by antisymmetry. A zero vector means the whole line is contained in the plane. Dually, the plane through the line and a point or ideal direction $D$ is given by the second formula; it vanishes if $D$ is already on the line. These operations require no Euclidean lengths or angles.

The Plücker embedding of $\mathrm{Gr}(2,{\mathbb{C}}^4)$ is a smooth projective quadric in ${\mathbb{CP}}^5$. An antisymmetric $4\times 4$ matrix is nonzero and decomposable precisely when its Pfaffian vanishes, giving the displayed equation. The wedge pairing of two points on this quadric vanishes exactly when the corresponding two-planes intersect. Thus the Klein quadric carries the complex conformal null-separation relation of complexified compactified Minkowski space.