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