Lagrange identity for the cross product

For three-dimensional vectors,The corresponding identity in ${\mathbb{R}}^n$ is

For real three-dimensional vectors, Lagrange's identity gives $|a\times b{|}^2=a^{T}Ma$ with symmetric matrix $M=|b{|}^{2}I-b{b}^T$. For $b\ne 0$, the normal line spanned by $b$ is its kernel and the perpendicular plane is its $|b{|}^2$ eigenspace. Consequently $M$ is positive semidefinite, has rank two and has zero determinant. For $b=0$, it is the zero matrix.