Sd  =  
        DiagonalMatrix[ 
          Sqrt[{e2^2 m3^2  +  
                e3^2 m2^2, 
e1^2 m3^2  +  
                e3^2 m1^2, 
e1^2 m2^2  +  
                e2^2 m1^2}]];
Fd = 
        DiagonalMatrix[ {e2 e3, e1 e3,  
            e1 e2}];
Nd = DiagonalMatrix[ {m2 m3, m1 m3,  m1 m2}];

xi = {ξ1, ξ2, ξ3}

NormSq[xi_] :=  
      xi[[1]]^2  +  xi[[2]]^2  +  xi[[3]]^2

Ddef = (NormSq[Sd . xi])^2  -  
        4 NormSq[Fd . xi] NormSq[Nd . xi]

Δ12  =  
        e1^2 m2^2  -  e2^2 m1^2;
Δ21  =  -Δ12;
Δ13  =  
        e1^2 m3^2  -  e3^2 m1^2;
Δ23  =  
        e2^2 m3^2  -  e3^2 m2^2;
Δ32  =  -Δ23;

Dexp1 = Δ23^2 ξ1^4  +  Δ13^2 ξ2^4  + Δ12^2 ξ3^4  + 
          2 Δ23 Δ13 ξ1^2 ξ2^2 - 
          2 Δ23 Δ12 ξ1^2 ξ3^2 + 
          2 Δ13 Δ12 ξ2^2 ξ3^2;
FullSimplify[Ddef - Dexp1]

Dexp2 = (Δ23 ξ1^2  +  Δ13 ξ2^2  + Δ12 ξ3^2)^2 - 
          4  Δ12 Δ23 ξ1^2 ξ3^2;
FullSimplify[Dexp2 - Ddef]

Dexp3 = (Δ23 ξ1^2  +  Δ13 ξ2^2  + Δ21 ξ3^2)^2 - 
          4  Δ21 Δ13 ξ2^2 ξ3^2;
FullSimplify[Dexp3 - Ddef]

Dexp4 = (Δ32 ξ1^2  +  Δ13 ξ2^2  + Δ12 ξ3^2)^2 - 
          4  Δ13 Δ32 ξ1^2 ξ2^2;
FullSimplify[Dexp4 - Ddef]

NotebookPrint[
      SelectedNotebook[], "/home/fdahl/writing/lis/notebooks/expressions \
for D.nb.ps"]