Quit

$Assumptions=θ∈Reals&&ϕ∈Reals&&ψ∈Reals;

$PrePrint=Replace[#,mat_?MatrixQ:>MatrixForm[mat]]&

(J[1]=-I{{0,1/2},{1/2,0}})//MatrixForm

(J[2]=-I{{0,-I/2},{I/2,0}})//MatrixForm

(J[3]=-I{{1/2,0},{0,-1/2}})//MatrixForm

LeviCivitaTensor[3][[1,2,3]]

Do[Print[{J[i].J[j]-J[j].J[i]-Sum[LeviCivitaTensor[3][[i,j,k]]J[k],{k,1,3}],i,j}],{i,1,3},{j,1,3}]

{x[1]=θ,x[2]=ϕ,x[3]=ψ}

{dx[1]=dθ,dx[2]=dϕ,dx[3]=dψ}

U=MatrixExp[x[2] J[3]].MatrixExp[x[1] J[1]].MatrixExp[x[3] J[3]]//Simplify

Simplify[Refine[ConjugateTranspose[U].U]]

Simplify[Det[U]]

UinvdU=Simplify[Sum[Inverse[U].D[U,x[k]]dx[k],{k,1,3}]]

Tr[J[1].J[1]]

σ[1]=FullSimplify[-2Tr[UinvdU.J[1]]]

σ[2]=FullSimplify[-2Tr[UinvdU.J[2]]]

σ[3]=FullSimplify[-2Tr[UinvdU.J[3]]]

As in Duff’s KK-SUGRA review (page 37)