You know all three components of e1 and you know e2z so e1.e2= e1xe2x+ e1ye2y+ e1ze2z= 0 is an equation with two unknown values, e2x and e2y. Similarly, e1.e3= e1xe3x+ e1ye3y+ e1ze3z= 0 is an equation with three unknown values, e3x, e3y, and e3z. The requirement that the vectors have unit length gives two more equations. So far that is 4 equations for the 5 unknown values. The requirement that e1Xe2= e3 gives the fifth equation.