Is there a way to factor out the middle tensor product? Given a state $(|1\rangle \otimes |0\rangle \otimes |1\rangle) + (|0\rangle \otimes |0\rangle \otimes |1\rangle)$, is it possible to factorise out the $|0\rangle$ in the middle of both of them?
Alternatively written as $|101\rangle + |001\rangle$ for convenience.
For a state such as the following it is possible to factorise out:
$|101\rangle + |100\rangle = |1\rangle (|01\rangle + |00\rangle)$
as shown above. Is there any way to factorise out the middle tensor in such a form?

$(|1\rangle \otimes |0\rangle \otimes |1\rangle) + (|0\rangle \otimes |0\rangle \otimes |1\rangle)$ is clearly a product state. It's tricky to find good notation to denote this and I don't think there's an established standard way. Here's one option:
Let $\sigma_{ij}$ be an automorphism that flips the $i$th and $j$th position of a tensor product $\bigotimes_{k=1}^n V$. Then
$$(|1\rangle \otimes |0\rangle \otimes |1\rangle) + (|0\rangle \otimes |0\rangle \otimes |1\rangle) = \sigma_{12} \Big( \ |0\rangle \ \otimes \ \big( |1\rangle \otimes |1\rangle + |0\rangle \otimes |1\rangle \big) \ \Big) $$
