What's the simplest characterisation of the quaternions? The one I'm most drawn to is Frobenius's theorem. It states that the quaternion algebra is the largest

What's the simplest characterisation of the quaternions? The one I'm most drawn to is Frobenius's theorem. It states that the quaternion algebra is the largest finite-dimensional associative division algebra over R. The computation needed to prove this is quite non-trivial (but ultimately fairly standard linear algebra) and it derives the relations i^2 = j^2 = k^2 = ijk = -1.