n=4

mat=Table[(Subscript[a, i,j]),{i,n},{j,n}]

{lu,p,c}=Simplify@LUDecomposition[mat]

MatrixForm[lu]

MatrixForm[l=LowerTriangularize[lu,-1]+IdentityMatrix[3]]

MatrixForm[u=UpperTriangularize[lu]]

Simplify[l.u]

The Swap

n=2;

A=B=IdentityMatrix[n];
Do[A=A.MatrixExp[SparseArray[{α,β}->(Subscript[ξ, α,β]),{n,n}]],{β,1,n},{α,1,β}]; (* This specifies the PBW ordering *)
Do[B=MatrixExp[SparseArray[{α,β}->(Subscript[η, α,β]),{n,n}]].B,{α,2,n},{β,1,α-1}];
MatrixForm/@{A,B}

mat=CF[
(A/.(Subscript[ξ, αβ__]):>(Subscript[ξ1, αβ])).(B/.(Subscript[η, αβ__]):>(Subscript[η1, αβ]))
]

mat//MatrixForm

Options[LUDecomposition]

LUDecomposition[mat, Pivoting -> False]

{lu,p,c}=Simplify@LUDecomposition[mat, Pivoting -> False]

lu//MatrixForm

MatrixForm[l=LowerTriangularize[lu,-1]+IdentityMatrix[n]]

MatrixForm[u=UpperTriangularize[lu]]

Simplify[l.u]

Simplify[l.u-mat]

The Swap with ϵ

n=2;

A=B=IdentityMatrix[n];
Do[A=A.MatrixExp[SparseArray[{α,β}->(Subscript[ξ, α,β]),{n,n}]],{β,1,n},{α,1,β}]; (* This specifies the PBW ordering *)
Do[B=MatrixExp[SparseArray[{α,β}->ϵ (Subscript[η, β,α]),{n,n}]].B,{α,2,n},{β,1,α-1}];
MatrixForm/@{A,B}

mat=CF[
(A/.(Subscript[ξ, αβ__]):>(Subscript[ξ1, αβ])).(B/.(Subscript[η, αβ__]):>(Subscript[η1, αβ]))
]

mat//MatrixForm

{LU,p,c}=Simplify@LUDecomposition[mat, Pivoting -> False]

LU//MatrixForm

MatrixForm[L=LowerTriangularize[LU,-1]+IdentityMatrix[n]]

MatrixForm[U=UpperTriangularize[LU]]

Simplify[L.U]

Simplify[L.U-mat]

Series[L, {ϵ,0,3}] // MatrixForm

Series[U, {ϵ,0,3}] // MatrixForm

l=Simplify[MatrixLog[L]]

MatrixLog[U]

LogReduce[ℰ_]:=ℰ/.c_*Log[a_]:>Log@Factor[(a^c)]//.Log[a_]+Log[b_]:>Log@Factor[a b];

u=Simplify[MatrixLog[U]//. {Log[a_ b_]:>Log[a]+Log[b], Log[(E^(a_))]:>a, Log[1/a_]:>-Log[a]}]

Tr[l.Table[(Subscript[y, α,β]),{α,n},{β,n}]]

Tr[u.Table[(Subscript[x, α,β]),{α,n},{β,n}]]

$k=0

(Subscript[Λ2𝔼, d_->r_])[Λ_]:=Module[{k},(Subscript[𝔼, d->r])@@Table[SeriesCoefficient[Λ,{ϵ,0,k}],{k,0,$k}]];

sw[i_,j_]:=(Subscript[Λ2𝔼, {i,j}->{i,j}])[Tr[l.Table[(Subscript[y, α,β])[i],{α,n},{β,n}]/ϵ+u.Table[(Subscript[x, β,α])[j],{α,n},{β,n}]]/.{(Subscript[ξ1, αβ__]):>(Subscript[ξ, αβ])[i], (Subscript[η1, αβ__]):>(Subscript[η, αβ])[j]}]

sw[3,4]

(Subscript[utm, 2])[1,2->3]

(Subscript[utm, 2])[1,2->3]//sw[3,4]