Some Identities in Algebras Obtained by the Cayley-Dickson Process

On the Consimilarity of Split Quaternions and Split Quaternion Matrices

“…Thus (10) is also a solution to (7) under (8). The independence of x 1 and x 2 can be seen from two simple facts that Rex 1 = 0 and Rex 2 = 0.…”

“…This equation has a nonzero solution for x if and only if det(δ (ja, jb)) = 0, which is equivalent to (8). In the present case, the general solution can be written as…”

“…We know that every associative finite-dimensional algebra A over an arbitrary K is isomorphic with a subalgebra of the matrix algebra M n (K) . So we could find a faithful representation for the algebra A in the matrix algebra M n (K), [8]. For the split quaternion algebra H S , the mapping:…”

“…Also, there are many studies on geometric and physical meaning of the split quaternions [16]- [17]. In [8], author studied equations ax = xb, ax = xb and x 2 = a, in algebras obtained by the Cayley-Dickson process. One of this algebras is algebra of split quaternions.…”

Linear Quaternionic Equations and Their Systems

“…The author of the paper [6] classifies solutions of the quaternionic equation ax + xb = c. The empty set, one or two points or a plane are obtained as the sets of its solutions. In the article [1] they consider equations of the forms ax = xb and ax =xb for some generalizations of quaternions and octonions. In [3] they solve by matrix method the general linear quaternionic equation of the form (12) and linear quaternionic systems of some type.…”