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Commutability of Tensor components , an ambigious situation

1. The problem statement, all variables and given/known data
I have been working on defining the transpose in a tensorial view using the kroneck delta tensor.


2. Relevant equations
I will use tensor notation

3. The attempt at a solution
Let Tjk be a 2nd level tensor:

(Tjk)TP = Tkj

My first attempt is:
δjkTjk δkj = Tkj

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

δjkδkj Tjk = δjkTkk

the latter expression becoming ambigious, I can replace either k and acquire:
δjkTjj = Tjk
or
δjkTjj = Tkj
 

Orodruin

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