Simplify bra-ket notation with kronecker product and kronecker sum

Question

I am taking a quantum informatics and communication course, this is the first time I have faced with Dirac's Bra-ket notation.

I have the following equation(Swap gate with 3 cnot):

First equation

$|\phi_2\rangle = |i\bigoplus(i\bigoplus k)\rangle\bigotimes|i\bigoplus k\rangle$

the textbook derived this simplier formula :

$|\phi_2\rangle=|k\rangle\bigotimes|i\bigoplus k\rangle$

I came to the same answer , so my question is the rules I applied are they logically correct?

Here is my method:

1. "Get rid of" the Kronecker sums by using the following property:

$i\bigoplus k = i\bigotimes I+I\bigotimes k$

but I also now $|i\rangle$ and $|k\rangle \in \{0,1\}$ so identity matrix(I) become the skalar 1(since $|k\rangle$ and $|i\rangle$ is also scalar)

$|i\bigotimes I + I \bigotimes (i\bigotimes k)\rangle\bigotimes|i \bigoplus k \rangle$=

=$|i\bigotimes (i\bigotimes k)\rangle\bigotimes |i\bigoplus k\rangle$=

since Kronecker product is associative

=$|(i\bigotimes i)\bigotimes k\rangle\bigotimes |i\bigoplus k\rangle$

I am not sure with the following, $i\bigotimes i = 1 $if i is scalar

and then $1 \bigotimes k$ = k?

=$|k\rangle\bigotimes|i\bigoplus k\rangle$

Thank you for your time!