Uses my GA30.m package to compute the values of the energy momentum tensor multivectors, and relate those to the conventional tensor description of the same.

<< GA30`;

Calculation of components of the energy momentum tensor T(a) = F a (F^(† ))ϵ/2

ClearAll[e1, e2, e3, i, vecE, vecH, one, f, fDagger, vecJ, vecM, j, jDagger, cross]
e1 = Vector[1,1] ;
e2 = Vector[1,2];
e3 = Vector[1,3];
i = e1 ** e2 ** e3;
one = Scalar[1];

vecE = (Subscript["E", 1]) e1 + (Subscript["E", 2]) e2 + (Subscript["E", 3]) e3 ;
vecH = (Subscript["H", 1]) e1 + (Subscript["H", 2]) e2 + (Subscript["H", 3]) e3 ;
vecJ = (Subscript["J", 1]) e1 + (Subscript["J", 2]) e2 + (Subscript["J", 3]) e3 ;
vecM = (Subscript["M", 1]) e1 + (Subscript["M", 2]) e2 + (Subscript["M", 3]) e3 ;
j = η( c ρ - vecJ ) + i ** ( c (Subscript[ρ, m]) - vecM );
jDagger = η( c ρ - vecJ ) - i ** ( c (Subscript[ρ, m]) - vecM );
cross[a_, b_] := -i ** (a ⋀ b);

ClearAll[v,vv]
(*t[v_] := ϵ ( vecE ** v ** vecE + vecH ** v ** vecH + i ** ( vecH ** v ** vecE - vecE ** v ** vecH))/2*)
f := vecE + i **  vecH (η);
fDagger := vecE - i **  vecH (η);
t[v_] :=(ϵ/2) ( f ** v ** fDagger)
vv = { t[one], t[e1], t[e2], t[e3] };

(*vv[[1]]  // TexForm
vv[[2]]
vv[[3]]
vv[[4]]
*)

Grid[ { Subscript["T", #],  " = ", vv[[# + 1]]}  &/@ (Range[4]-1)]

Same calculation for the adjoint of the energy momentum tensor \overbar{T}(a) = (F^(† ))a F ϵ/2

ClearAll[t2,vv2]
t2[v_] := (ϵ/2) ( (vecE - i **  vecH (η)) ** v ** (vecE + i **  vecH (η)))
vv2 = { t2[one], t2[e1], t2[e2], t2[e3] };

Grid[ { Subscript["T", #],  " = ", vv2[[# + 1]]}  &/@ (Range[4]-1)]
Grid[ { Subscript["T", #],  " = ", vv[[#+1]] - vv2[[# + 1]]}  &/@ (Range[4]-1)]

tt = (ϵ/2)(fDagger ** j + jDagger ** f ) 
tt2 = - ϵ η ( ⟨vecE ,vecJ⟩ + ⟨vecH , vecM⟩ ) + ϵ c η ( ρ vecE + (Subscript[ρ, m])  vecH ) + ϵ cross[vecE, vecM] + ϵ (η^2) cross[ vecJ, vecH ]
tt - tt2
Simplify[tt2 /. {η -> Sqrt[μ/ϵ], c -> Sqrt[1/(μ ϵ)]}, Assumptions-> μ>0 && ϵ>0]