Rotation invariant tensors, isotropic tensors, follow up question A previous question established that the only rank 2 and 3 isotropic tensors are the $\lambda \delta_{ij}$ and the $\lambda \epsilon_{ijk}$ tensors respectively.  How do we know ... "All isotropic tensors of higher rank are obtained by combining $\delta_{ij}$ and $\epsilon_{ijk}$ tensor products, contractions, and linear combinations." IA Vector Calculus  That is, how do we know that, for example, every rank 4 isotropic tensor has the form  $\alpha \delta_{ij} \delta_{kl} + \beta \delta_{ik} \delta_{jl}+ \gamma \delta_{il} \delta_{jk}$
Note:  I'm pretty sure we can show, using rotations as in the link, that the only non-zero terms in an isotropic rank 4 tensor have the form $T{iijj},T{ijij},T{ijji}$ and that $T{ijij} = T{ikik}$ etc, so the terms in T are the ones appearing in form above, so ....
 A: I was looking for a quick easy answer.  This answer is easy but not quick.
By using rotations around the 3 axes of 90 and 180 deg. we can show that the non-zero components of the isotropic rank 4 tensor T have the form  $T{iijj},T{ijij},T{ijji}$ ( as the others, e.g $T_{1112} = -T_{1112} = 0$) and that $T{iijj} = T{iikk}$, $T{ijij} = T{ikik}$, $T{ijji} = T{ikki}$.
So, if T has a term $T_{1122}$ then T also has terms $T_{1133},T_{2211}, T_{2233}, T_{3311}$ and $T_{3322}$ and the terms are equal, so T contains six equal terms $\lambda = T{iijj}, i \neq j$
Similarly if T has a term $T_{1212}$ then T contains six equal terms  $\beta = T{ijij}, i \neq j$
and if T has a term $T_{1221}$ then T contains six equal terms  $\gamma = T{ijji}, i \neq j$
So, the possible terms in T are $T_{1111} = T_{2222} = T_{3333}$, $\lambda = T{iijj}$, $\beta = T{ijij}$, and $\gamma = T{ijji}$ with $i \neq j$.
Let T1 = $\lambda \delta_{ij}\delta_{kl}$, T2 = $\beta \delta_{ik}\delta_{jl}$, and T3 = $\gamma \delta_{il}\delta_{jk}$
A = T - T1 - T2 - T3 is isotropic as all its terms are isotropic, and only contains 3 terms, $A_{1111}, A_{2222}, A_{3333}$ and since A is isotropic all these terms must be 0 so that
T = T1 + T2 + T3.
