  Given a linear unknown system with $m$ inputs, $p$ outputs, $n$ dimensional
state vector, and $q$ dimensional ecosystem, the problem of the adaptive
optimal output regulation of this system boils down to iteratively solving a
set of linear equations and each of these equations contains $\frac{n (n+1)}{2}
+ (m+q)n$ unknown variables. In this paper, we refine the existing algorithm by
decoupling each of these linear equations into two lower-dimensional linear
equations. The first one contains $nq$ unknown variables, and the second one
contains $\frac{n (n+1)}{2} + mn$ unknown variables. As a result, the
solvability conditions for these equations are also significantly weakened.
