Let $k>0$ be a positive integer and $n=4k.$  A special case of a result of Dieudonn\'e  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 $G.$

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 $g_i$ is also doubly stochastic.