The real quaternion algebra is studied using the fact that a real quaternion \(a\) has one complex and one real representation. It is shown that for any \(a\) over the real quaternion algebra there are two independent unitary matrices such that \(a\) satisfies two universal similarity factorizations. Then various properties of quaternions derived from these universal similarity factorizations are presented. It is also shown that all systems of linear equations over the real quaternion algebra can be solved by transforming them into conventional systems of linear equations over the real field.