Commutability of Tensor components , an ambigious situation

[Somali_Physicist]

Homework Statement

I have been working on defining the transpose in a tensorial view using the kroneck delta tensor.

Homework Equations

I will use tensor notation

The Attempt at a Solution

Let T_jk be a 2^nd level tensor:

(T_jk)^TP = T_kj

My first attempt is:
δ_jkT_jk δ_kj = T_kj

However if tensor components are commutable by nature the above expression can be

δ_jkδ_kj T_jk = δ_jkT_kk

the latter expression becoming ambigious, I can replace either k and acquire:
δ_jkT_jj = T_jk
or

δ_jkT_jj = T_kj​

 

[Orodruin]

δjkTjk δkj = Tkj

Please read https://www.physicsforums.com/insights/the-10-commandments-of-index-expressions-and-tensor-calculus/
In particular commandment 5, which is typeset in boldface.