Are $SL(n, \mathbb{R})$ and $SO(n,\mathbb{R})$ the same group?

Question

The special linear group is the set of automorphisms with determinant $1$.

The special orthogonal group is the set of orthogonal automorphisms with determinant $1$.

The special orthogonal group is included in the special linear group, is this inclusion a strict inclusion? Are there automorphisms with determinant $1$ which are not orthogonal?

(I ask the question in the frame of a finite dimensional vector space.)

Answer 1

Yes of course, for example the matrix

$$A=\begin{pmatrix}1&1\\0&1\end{pmatrix}\in\operatorname{SL}_2(\Bbb R)$$ but $A\not\in\mathcal O_2(\Bbb R)$ since $AA^T\ne I_2$.