Problem with definition of the generalized delta

Question

EDIT: ealier I gave bad example, so let me again write post with proper ones.

I find quite problematic using generalized delta. For example:

{a,b,c,d,m,n,s}::Indices.
{a,b,c,d,m,n,s}::Integer(1..4).
\delta{#}::KroneckerDelta.

\epsilon_{a b c d}::EpsilonTensor(delta=\delta).

delta1:=\delta_{a b c d};
expand_delta(_);

delta2:=\delta_{a c b d};
expand_delta(_);

delta3:=\delta^{a b}_{c d};
expand_delta(_);

gives

I would expect that \delta_{abcd} would give me middle outcome. What I'm missing?

Ps. I use cloud cadabra.

Answer 1

The help for KroneckerDelta states that indices are 'paired' in the way it comes out here. It is perhaps more transparent if you write upper and lower indices, so

$$2 \delta_{a}{}^{b}{}_{c}{}^{d} = \delta_{a}{}^{b} \delta_{c}{}^{d} - \delta_{c}{}^{b} \delta_{a}{}^{d}.$$

Your version just has all indices moved to subscript position.

It's all a matter of convention of course (there's nothing wrong with defining things 'your' way), but this is the convention Cadabra uses.