Some Authors, e.g. Baez, Ward, defined multiplication of octonions by formula $ (a,b) \cdot^B (c,d)=(ac-db^*, cb+a^*d) \textrm{ for } a,b,c,d\in \mathbb H, $ some anothers, e.g. Springer&Veldkamp, N. Jacobson, by $ (a,b)\cdot^S(c,d)=(ac-d^*b, da+bc^*) \textrm{ for } a,b,c,d\in \mathbb H $ ($a^*$ denotes the conjugate quaternion to $a$).

Which of these mltiplications is better? Are algebras $\mathbb O=\mathbb H \times \mathbb H$ with multiplications $\cdot^B$ and $\cdot^S$ (and the standard addition and multiplications by real numbers) isomorphic?

Some authors, e.g. Baez, Ward, defined multiplication of octonions by formula $ (a,b) \cdot^B (c,d)=(ac-db^*, cb+a^*d) \textrm{ for } a,b,c,d\in \mathbb H, $ some others, e.g. Springer & Veldkamp, N. Jacobson, by $ (a,b)\cdot^S(c,d)=(ac-d^*b, da+bc^*) \textrm{ for } a,b,c,d\in \mathbb H $ ($a^*$ denotes the conjugate quaternion to $a$).

Which of these multiplications is better?

Are algebras $\mathbb O=\mathbb H \times \mathbb H$ with multiplications $\cdot^B$ and $\cdot^S$ (and the standard addition and multiplications by real numbers) isomorphic?