What is the geometrical interpretation of this vector identity (Binet-Cauchy identity)?

Question

Sometimes I use this identity really useful to solve the problem, $$\mathbf{\left(A\times B\right)\cdot}\left(\mathbf{C}\times\mathbf{D}\right)=\left(\mathbf{A}\cdot\mathbf{C}\right)\left(\mathbf{B}\cdot\mathbf{D}\right)-\left(\mathbf{B}\cdot\mathbf{C}\right)\left(\mathbf{A}\cdot\mathbf{D}\right)$$ I know this is derived from Binet-Cauchy identity, $$\biggl(\sum_{i=1}^n a_i c_i\biggr) \biggl(\sum_{j=1}^n b_j d_j\biggr) = \biggl(\sum_{i=1}^n a_i d_i\biggr) \biggl(\sum_{j=1}^n b_j c_j\biggr) + \sum_{1\le i < j \le n} (a_i b_j - a_j b_i ) (c_i d_j - c_j d_i )$$ But what is the geometrical meaning of this identity? Can anyone explain this without using too many equations?