TITLE: A question about special linear group
QUESTION [8 upvotes]: Is there any way to find all matrices $G \in SL(n,\mathbb Z)$ such that there exists a matrix $A \in GL(n,\mathbb R)$ satisfying
$$
AGA^{-1} \in SO(n,\mathbb R)?
$$

REPLY [14 votes]: These are exactly the finite-order elements:

If $G \in \text{SL}(n,\mathbb{Z})$ has finite order, then there exists an inner product preserved by $G$ (take an arbitrary inner product and add up its images under all powers of $G$).  Changing bases to an orthonormal basis for this invariant inner product has the effect of conjugating $G$ into the orthogonal group.
Conversely, if $G \in \text{SL}(n,\mathbb{Z})$ is such that there exists some $A \in \text{GL}(n,\mathbb{R})$ with $A G A^{-1} \in \text{SO}(n,\mathbb{R})$, then since $A \cdot \text{SL}(n,\mathbb{Z}) \cdot A^{-1}$ is a discrete subgroup of $\text{GL}(n,\mathbb{R})$, its intersection with the compact group $\text{SO}(n,\mathbb{R})$ is a finite group, and thus $G$ has finite order.