Let $O(n)$ and $SO(n)$ denote the split orthogonal linear algebraic group and its special subgroup, over some fixed field of characteristic not two. I am looking for a reference that explains how to describe the simple (finite-dimensional) representations of $O(n)$ in terms of the simple representations of $SO(n)$.

The relation between the complex representations of the corresponding compact Lie groups is explained in section VI.7 of Bröcker and tom Dieck’s Representations of Compact Lie Groups. I believe the key statements made there also hold in the above algebraic situation, and I have worked this out in a fair amount of detail, but I am currently unwilling to believe that this has not already been done somewhere in the literature. Roughly, these statements are as follows:

• When $n$ is odd, $O(n)$ is a direct product of $SO(n)$ with $\mathbb Z/2$. Thus, every simple $SO(n)$-representation can be lifted to two distinct $O(n)$-representations, and every $O(n)$-representation arises in this way. (See also this question regarding simple representations of products.)

• When $n$ is even, $O(n)$ is only a semi-direct product of $SO(n)$ with $\mathbb Z/2$. In this case, only some simple $SO(n)$-representations can be lifted. Those that can be lifted can again be lifted to two distinct $O(n)$-representations. The remaining simple $SO(n)$-representations occur in pairs whose direct sum can be lifted to a unique simple $O(n)$-representation. All simple $O(n)$-representations arise in either of these ways.

Of course, in this question I am mainly interested in references concerning the case when $n$ is even.