Corollary 1

Let A be a central simple K-algebra of finite degree $>1$ whose elements are all algebraic of degree $\leqslant 2$ over K. Then A is isomorphic to a quaternion algebra over K.

If K is finite, then the algebra A is isomorphic to a matrix algebra $\mathbf{M}_n(K)$ (VIII, p. 357, Corollary 2) and therefore contains elements of degree $n$ over K; the assumption implies $n= 2$ and therefore the result in this case (VIII, p. 362, Example). Suppose that the field K is infinite. Let L be a maximal étale subalgebra of A. By V, §7, No. 4, p. 41, Proposition 7, there exists an element $x$ of A such that the K-algebra L is equal to $K[x]$, hence by assumption has degree $\leqslant 2$. Since we have $[A : K] = [L : K]^2$ (VIII, p. 264, Proposition 4 and p. 262, Proposition 3), we conclude that [A : K] = 4. Corollary 1 then follows from Proposition 4.