Quotient Group of Quadratic Residues Modulo 2 of 2-adic Units

Theorem
Let $\Z_2$ be the $2$-adic integers.

Let $\Z_2^\times$ denote the set of $2$-adic units.

Let $\paren{\Z_2^\times}^2 = \set{a^2 : a \in \Z_2^\times}$

Then the multiplicative quotient group $\Z_2^\times / \paren{\Z_2^\times}^2$ has order $8$ with:
 * $\set{1, -1, 5, -5, 2, -2, 10, -10}$ as a transversal