On the theorems of Frobenius, Zorn and Hopf’s commutativity

Let \(A\) be a real algebra. It is called <i> locally complex algebra</i> if every non-zero element generates a subalgebra isomorphic to either \(\mathbb{R}\) or \(\mathbb{C}.\) It is said to satisfy the <em>uniqueness of the square root except the sign</em> if the equation \(x^2=y^2\) implies \(y=\pm x.\) We show the following: 1. Every locally complex algebra is a quadratic algebra. 2. Every alternative locally complex algebra is isomorphic to either \(\mathbb{R},\) \(\mathbb{C},\) \(\mathbb{H}\) or \(\mathbb{O}.\) 3. Every commutative locally complex algebra without divisors of zero is isomorphic to either \(\mathbb{R}\) or \(\mathbb{C}.\) 4. Every finite-dimensional algebra satisfying the uniqueness of the square root except the sign has dimension \(\leq 2\) and contains non-zero idempotents.