Let $f:\mathbb{R}^3 \to \mathbb{R}^3: x \mapsto (a_1\times x+a_2)\times(b_1 \times x+b_2)$ with $a_1,a_2,b_1,b_2\in\mathbb{R}^3$. Here, $\times$ denotes the standard cross product on $\mathbb{R}^3$.
I want to prove the following statement: The set of roots $R=\{x\in\mathbb{R}^3:f(x)=0\}$ of $f$ is a finite union of affine subspaces of $\mathbb{R}^3$ for any $a_1,a_2,b_1,b_2$.
Mathematica tells me that, generically, $R$ is either a union of two lines, or it is empty. There are some other cases, like $R=\mathbb{R}^3$ (by choosing $b_1=\lambda a_1$ and $b_2=\lambda a_2$), or $R=\mathrm{span}\{a_1,b_1\}$ (by choosing $a_2=0=b_2$).
Specifically, I would be interested in the following questions:
- The system $f(x)=0$ has three quadratic equations in three unknowns. Generically, one would expect the roots to be a finite set of points. Is it somehow geometrically obvious from the definition of $f$ that the roots are never isolated?
- Can the statement be proved using mostly properties of the cross product? Or does one need algebraic geometry?
- Given any system of algebraic equations, is there is a generic way to see whether its roots are a finite union of affine spaces? Is there anything special or interesting about algebraic equations with this property?