Four-vector scalar triple product identity ($[b,c,d]a-[a,c,d]b+[a,b,d]c-[a,b,c]d=0$)

For four vectors in three-dimensional real Euclidean space, write $[a,b,c]=a\cdot (b\times c)$. Expand the same cross product in two ways using the vector triple product identity:Rearrangement gives the displayed identity. It expresses linear dependence of four vectors in three dimensions with explicit determinant coefficients and remains valid when some or all of those coefficients vanish.