Take the dot product of equation (*) with … to prove …

Take the dot product of equation (*) with $\mathbf{D}$ to prove
$$(\mathbf{A}\times \mathbf{B})\cdot (\mathbf{C}\times \mathbf{D})=(\mathbf{A}\cdot \mathbf{C})(\mathbf{B}\cdot \mathbf{D})-(\mathbf{B}\cdot \mathbf{C})(\mathbf{A}\cdot \mathbf{D}).$$

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §11.3 Cross Products and Determinants, #48