We say that a linear form $f=ax+by+cz$ of real coefficients is "irrational" if $a,b,c$ are linearly independent over the rationals. My question is: Are there 3 such "irrational" linear forms $f_1,f_2,f_3$ such that the product $f_1f_2f_3$ is of integral coefficients?

Note that for the "2-dimensional" analogue, we have $(x+\sqrt{2}y)(x-\sqrt{2}y)=x^2-2y^2$.