Quaternions Without Imaginarities or the Vector Representation of Quaternions

The general quaternionic algebraic structure we are considering was provided earlier by the author with a commutative product and will be provided here with a non-commutative product. We replace the imaginary units usually used in the theory of quaternions by linearly independent vectors and the usual Hamilton product rule by a Hamiltonian-adapted vector-valued vector product and prove both a new geometric property of this product and a vectorial adopted Euler type formula.