A somewhat random seeming complex exponential evaluation using CliffordBasic, and an R3 bivector argument.

<< CliffordBasic`;
$SetSignature={3,0};

ClearAll[ a, b,c, angle, exp, r ]
exp[ t_ ] := Module[ {abs},
abs = Sqrt[-GeometricProduct[t,t]];
Cos[ abs ] + (t/abs) Sin[ abs ]];
r[t_] := GeometricProduct[ e[3], exp[t ] ];
a =  r[e[3,1] Pi/4]
b  =  r[e[3,2] Pi/4]
c = GeometricProduct[ a,b] // Simplify
j = (e[1,2]+e[1,3]-e[2,3])/Sqrt[3];
exp[ j Pi/3] - c // Simplify
(*angle = exp[ e[1,2] Pi(-1/3 + 1/6)]
iangle = MultivectorInverse[angle]// Simplify
GeometricProduct[ c,iangle ]*)