Prove or disprove: Special linear group is subgroup of orthogonal group

Question

Let $H$ be the special linear group. $H=\{M \in M_n(\mathbb{K}): det(M) = 1\}$. Is $H$ a subgroup of the orthogonal group?

Counterexample:

Let $A = \left( \begin{array}{ccc} 1 & 1 \\ 0 & 1 \\ \end{array} \right)$

Then $det(A) = 1$. But $AA^T \neq Id$. So $A$ isn't orthonormal.

Is that a valid counterexample?