Dieudonné and generators of the orthogonal group

Question

Let $k>0$ be a positive integer and $n=4k.$ A special case of a result of Dieudonné is that every element $g$ of the orthogonal group in $n$ variables over the rational numbers $$ G=O(n,\mathbb{Q}) $$ is a product of at most $n$ elements of order $2$ of $G$.

We want to add some extra condition on $g$ (e.g., the following one)

Question: Can the result be extended to the subgroup of doubly stochastic elements $g$ of $G$.

More precisely:
Let $g \in G$ be such that $g$ is doubly stochastic, that is, the sum of the entries in any fixed row (or column) of $g$ always equal $1$. Assume that $g$ has order $>2$ (e.g., $g$ has infinite order) in $G.$ Note also that the identity has order $1$ by hypothesis.

There exists a decomposition

$$ g =g_1 g_2 \cdots g_n $$ with $g_i \in G$ such that

(a) either $g_i$ is equal to the identity or $g_i$ has order $2.$

and

(b) at least one of the $g_i,$ that is not equal to the identity, is also doubly stochastic.