Question about properites of tensor product

[td21]

They are being 2 by 2 matrices and I being the identity. Physically they are Pauli matrices.

1. Is $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes B$$

= $$(A\otimes I\otimes I)\otimes B + (I\otimes A\otimes I)\otimes B + (I\otimes I\otimes A)\otimes B$$? I think so.

2. Is $$(A\otimes I\otimes I)\otimes B + (I\otimes A\otimes I)\otimes B + (I\otimes I\otimes A)\otimes B$$=

$$(A\otimes B)\otimes (I\otimes I)\otimes (I\otimes I) + (I\otimes I)\otimes (A\otimes B)\otimes (I\otimes I) + (I\otimes I)\otimes(I\otimes I)\otimes(A\otimes B)$$? I am not sure.

Note that they are no scalar product here. I ask 2 because I stumbled upon the RHS of 2 and hope to know if it can be factored out as the LHS of 1. So basically I reverse the line of reason to ask these two questions. Looking at the LHS of 1., I also want to know:

3. Is $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes B$$
= $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes (B\otimes I\otimes I)$$
= $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes (I\otimes B\otimes I)$$
=$$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes (I\otimes I\otimes B)$$? I do not think so as I do not think I can enlarge the dimension of B here?
This is not homework.

 

[andrewkirk]

1 is correct.
2 is not, because the terms on the LHS are order-4 tensors and those on the RHS are order-6 tensors. If the atomic elements all have the same order then each term has to have the same number of $\otimes$ symbols in it.
3 is not correct, for the same reason.