zero=IdentityMatrix[4]*0

s1=zero; s1[[3,4]]=-1; s1[[4,3]]=1;s1//MatrixForm

s2=zero; s2[[2,4]]=1; s2[[4,2]]=-1;s2//MatrixForm

s3=zero; s3[[2,3]]=-1; s3[[3,2]]=1;s3//MatrixForm

k1=zero; k1[[1,2]]=1; k1[[2,1]]=1;k1//MatrixForm

k2=zero; k2[[1,3]]=1; k2[[3,1]]=1;k2//MatrixForm

k3=zero; k3[[1,4]]=1; k3[[4,1]]=1;k3//MatrixForm

LorentzTransformationMatrix[RotVec_,BoostVec_]:=(Lmat=-RotVec[[1]]s1-RotVec[[2]]s2-RotVec[[3]]s3-BoostVec[[1]]k1-BoostVec[[2]]k2-BoostVec[[3]]k3; MatrixExp[Lmat])

GeneralTransf=LorentzTransformationMatrix[{r1,r2,r3},{b1,b2,b3}];GeneralTransf//MatrixForm

Re[GeneralTransf]/.{r1->1.0/(Sqrt[5]),r2->(Sqrt[(1/5)]),r3->(Sqrt[(3/5)]),b1->5.0,b2->7.0,b3->9.0}//MatrixForm

LorentzTransformationMatrix[{0,Pi,0},{0,0,0}]//MatrixForm  (* example of rotation by Pi about the y-axis  *)

LorentzTransformationMatrix[{0,0,0},{z1,0,0}]//FullSimplify//MatrixForm  (* example of a boost along the x-axis with rapidity z1->(Subscript[ζ, 1]) *)

Field Transformations:

Λ={{γ,-β γ,0,0},{-β γ,γ,0,0},{0,0,1,0},{0,0,0,1}}; MatrixForm[Λ]

Fmat={{0,-Ex,-Ey,-Ez},{Ex,0,-Bz,By},{Ey,Bz,0,-Bx},{Ez,-By,Bx,0}}; MatrixForm[Fmat]

FmatPrime=Λ. Fmat . Transpose[Λ] //Simplify; FmatPrime//MatrixForm

MatrixForm[FmatPrime]/.(β^2)->-(1/(γ^2))+1

Λ1=Λ;  Λ1//MatrixForm

(({{γ, -β γ, 0, 0}, {-β γ, γ, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}))

Λ2= (({{γ, β γ, 0, 0}, {β γ, γ, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}))

Λ1.Λ2//Simplify//MatrixForm

epsilon=LeviCivitaTensor[4,List];

epsilon[[1,2,3,4]]

epsilon[[2,1,3,4]]