The set of all quaternions is a vector space over the real numbers with dimension 4. Multiplication of quaternions is associative and distributes over vector addition, but with the exception of the scalar subset, it is not commutative. Therefore, the quaternions are a non-commutative, associative algebra over the real numbers. Even though contains copies of the complex numbers, it is not an associative algebra over the complex numbers.
tHE sET oF all qUateRNIonS IS a vectoR Space oVEr thE reaL NumbERs WItH dimension 4 multIplICatIon of qUAtErNioNs is assoCIaTive And dISTributeS oVEr vEcTor aDditIoN bUt With the excEptiOn oF the scalar subset it is nOt COmmUtATIVe tHereFOrE THe QUaTerNions are A non-CommUTATiVE aSsOcIative algebrA over The reaL numBers even thougH cOntAins copiEs of THe COMpLEx numbErS it iS not an asSOciAtivE algEBra oveR thE comPleX numBers