  Let $(R,*)$ be a ring with involution and let $A = {\rm M}(n,R)$ be the
matrix ring endowed with the $*$-transpose involution. We study ${\rm
SL}_*(2,A)$ and the question of Bruhat generation over commutative and
non-commutative local and ad\`elic rings $R$. An important tool is the property
of a ring being $*$-Euclidean. In this regard, we introduce the notion of a
$*$-local ring $R$, prove that $A$ is $*$-Euclidean and explore reduction
modulo the Jacobson radical for such rings. Globally, we provide an affirmative
answer to the question wether a commutative ad\`elic ring $R$ leads towards the
ring $A$ being $*$-Euclidean; while the non-commutative ad\`elic quaternions
are such that $A$ is $*$-Euclidean and ${\rm SL}_*$ is generated by its Bruhat
elements if and only if the characteristic is $2$.

Group Theory