In this paper, \thinspace \thinspace we generalize the concepts of \thinspace level \thinspace and \thinspace sublevel of a composition algebra to algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every}$t\in \Bbb{ N},${ \small % \ a division algebra}$A_{ t}${ \small \ of dimension}$2^{ t}${ \small \ over the power-series field}$K\{ X_{ 1},X_{ 2},...,X_{ t}\}.\,${ \small \ This gives us the possibility to construct a division algebra of \thinspace dimension \thinspace 2}$^{ t}\,${ \small and prescribed \thinspace level and sublevel \thinspace \thinspace 2}$^{ k}${ \small ,\thinspace \thinspace}$k,\,t\in \Bbb{ % N}^{ *}${ \small \ and a division algebra of \thinspace dimension \thinspace}$% 2^{ t}+1,t\in \Bbb{ N}\,${ \small and prescribed \thinspace level and sublevel}% $\,2^{ k}+1,k\in \Bbb{ N}.\,\medskip $
Categories: math.RA