I'm reading a group on algebraic groups (the one by Meinolf Geck), and it defines the orthogonal group as the group of $ n x n$ matrices $A$ such that $A^{tr}QA=Q$ where $Q$ is such that $Q_{ij}=1$ for $i+j=n+1$ and $Q_{ij}=0$ otherwise, i.e. $Q$ has ones on the "lower-left to upper-right diagonal" and zeros elsewhere. It seems to me that $Q$ should be the identity matrix, and that this doesn't coincide with the definition I'm familiar with. Below is a copy of the text. What's going on here?
The text defines $\Gamma(Q,k)$ to be those matrices $A$ (with entries in a field $k$) such that $A^{tr}QA=Q$.