  Bowman and Bradley obtained a remarkable formula among multiple zeta values.
The formula states that the sum of multiple zeta values for indices which
consist of the shuffle of two kinds of the strings $\{1,3,\ldots,1,3\}$ and
$\{2,\ldots,2\}$ is a rational multiple of a power of $\pi^2$. Recently, Saito
and Wakabayashi proved that analogous but more general sums of finite multiple
zeta values in an adelic ring $\mathcal{A}_1$ vanish. In this paper, we
partially lift Saito-Wakabayashi's theorem from $\mathcal{A}_1$ to
$\mathcal{A}_2$. Our result states that a Bowman-Bradley type sum of finite
multiple zeta values in $\mathcal{A}_2$ is a rational multiple of a special
element and this is closer to the original Bowman-Bradley theorem.
