For certain non-associative real Banach algebras the author obtains a Gelfand-Mazur type theorem (each division algebra of the considered class is isomorphic to either of \({\mathbb{R}}\), \({\mathbb{C}}\), \({\mathbb{H}}\) (the quaternions), \({\mathbb{D}}\) (the Kelley numbers)), and in the commutative case a Shilov type theorem (in the considered class each algebra without topological divisors of zero is isomorphic either to \({\mathbb{R}}\) or to \({\mathbb{C}})\).