I came across this factorization:
$$ \begin{align} \frac12 (a + c)(b - d) + \frac12 (a - c)(b + d) &= \frac12(ab-ad+cb-cd+ab+ad-cb-cd) \\[4pt] &= \frac12(2ab - 2cd) \\[4pt] &= ab - cd \end{align} $$
Is there a technical mathematical name for this? I've seen it briefly being referred to as "twisted factorization" but it doesn't seem prevalent.
On
Not sure about the name, but this follows from the factorization of matrices: $$\left[\begin{array}{cc}\frac{a+c}2&\frac{b+d}2\\-(a-c)&b-d\end{array}\right]=\left[\begin{array}{cc}\frac 12&\frac 12\\ -1&1\end{array}\right]\left[\begin{array}{cc}a&d\\ c&b\end{array}\right].$$ Geometrically, this can be explained as follows: Let $v_1=<a,d>,v_2=<c,b>$ be two vectors. Then $$v_1\wedge v_2=\frac 12(v_1+v_2)\wedge (v_2-v_1),$$ i.e. the area of the parallelogram spanned by $v_1,v_2$ is half of the area of the parallelogram spanned by the vectors representing the diagonals of the previous parallelogram.
One way to view this identity is as saying that $A = \begin{pmatrix}\frac1{\sqrt2} & \frac1{\sqrt2} \\ -\frac1{\sqrt2} & \frac1{\sqrt2} \end{pmatrix}$ defines a unitary operator on $\mathbb R^2$, for the standard inner product. Since, when $v = (a, -c)^t$ and $w = (b, d)^t$, the LHS expresses $\langle Av, Aw \rangle$ and the RHS expresses $\langle v, w \rangle$.