Off[General::"spell1"]

f=(({{0, ex, ey, ez}, {-ex, 0, bz, -by}, {-ey, -bz, 0, bx}, {-ez, by, -bx, 0}}));  (* let's use units where c=1 *)

lambda=(({{γ, -γ β, 0, 0}, {-γ β, γ, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}}));

(* Now to construct the transformed tensor *)

fp00=Sum[ lambda[[1,λ]] *lambda[[1,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp01=Sum[ lambda[[1,λ]] *lambda[[2,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp02=Sum[ lambda[[1,λ]] *lambda[[3,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp03=Sum[ lambda[[1,λ]] *lambda[[4,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp10=Sum[ lambda[[2,λ]] *lambda[[1,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp11=Sum[ lambda[[2,λ]] *lambda[[2,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp12=Sum[ lambda[[2,λ]] *lambda[[3,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp13=Sum[ lambda[[2,λ]] *lambda[[4,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp20=Sum[ lambda[[3,λ]] *lambda[[1,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp21=Sum[ lambda[[3,λ]] *lambda[[2,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp22=Sum[ lambda[[3,λ]] *lambda[[3,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp23=Sum[ lambda[[3,λ]] *lambda[[4,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp30=Sum[ lambda[[4,λ]] *lambda[[1,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp31=Sum[ lambda[[4,λ]] *lambda[[2,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp32=Sum[ lambda[[4,λ]] *lambda[[3,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fp33=Sum[ lambda[[4,λ]] *lambda[[4,σ]]*f[[λ,σ]]    ,{λ,1,4},{σ,1,4}]

fpmatrix={{fp00,fp01,fp02,fp03},{fp10,fp11,fp12,fp13},{fp20,fp21,fp22,fp23},{fp30,fp31,fp32,fp33}};

MatrixForm[%]

(* Now to compare the result of the transformation against Eq. 12.108 *)

exprime=fpmatrix[[1,2]] //Simplify  (* this equals ex, since (γ^2)(1-(β^2)) = 1. Transform holds! *)

eyprime=fpmatrix[[1,3]] //Simplify  (* Transform holds! *)

ezprime=fpmatrix[[1,4]]  //Simplify (* Transform holds! *)

bxprime=fpmatrix[[3,4]] //Simplify (* Transform holds! *)

byprime=fpmatrix[[4,2]]  //Simplify(* Transform holds! *)

bzprime=fpmatrix[[2,3]] //Simplify (* Transform holds! *)