This is kind of overkill but here goes:
You can also do this using norms. has a norm .
Suppose , then .
So we're looking to solve .
Now since for the last equality.
So as you can see to find a solution, we're left to solve a form of Pell's equation, which is solvable i.e. we can find and . Thus has an inverse.
After Pell's equation is solved though, we'll have an answer in terms of and , so this proof needs some polishing at the end. I won't do this however since I've already given you another solution; this posting was just a thought I had.