d2=Table[0, {α, 3}, {β, 3}];
bigD2=Table[0, {α, 3}, {β, 3}];
i2=Table[KroneckerDelta[α,β], {α, 3}, {β, 3}];
d2//MatrixForm
bigD2//MatrixForm
i2//MatrixForm

bigD2 = d2-(1/3)*Tr[d2]*i2

Table[bigD2[[α,β]]=d2[[α,β]]-(1)*Sum[d2[[α,μ]],{μ,1,3}]*KroneckerDelta[α,β]-(1)*Sum[d2[[ν,β]],{ν,1,3}]*KroneckerDelta[β,α], {α, 3}, {β, 3}]

d3=Table[0, {α, 3}, {β, 3}, {γ, 3}]
bigD3=Table[0, {α, 3}, {β, 3}, {γ, 3}]
i3=Table[KroneckerDelta[α,β,γ], {α, 3}, {β, 3}, {γ, 3}]
d3//MatrixForm
bigD3//MatrixForm
i3//MatrixForm

bigD3=d3-(1/3)*Tr[d3]*i3

Table[bigD3[[α,β,γ]] = d3[[α,β,γ]]-(1/3)*Sum[d3[[μ,μ,μ]],{μ,1,3}]*KroneckerDelta[α,β,γ], {α, 3}, {β, 3}, {γ, 3}]

Table[bigD3[[α,β,γ]] = d3[[α,β,γ]]-(1/5)*Sum[d3[[α,μ,μ]],{μ,1,3}]*KroneckerDelta[β,γ]-(1/5)*Sum[d3[[ν,β,ν]],{ν,1,3}]*KroneckerDelta[γ,α]-(1/5)*Sum[d3[[ρ,ρ,γ]],{ρ,1,3}]*KroneckerDelta[α,β], {α, 3}, {β, 3}, {γ, 3}]