They don't remain exactly the same. The answers to this (https://math.stackexchange.com/questions/641809/what-specific-algebraic-properties-are-broken-at-each-c

They don't remain exactly the same. The answers to this (https://math.stackexchange.com/questions/641809/what-specific-algebraic-properties-are-broken-at-each-cayley-dickson-stage-beyon https://math.stackexchange.com/questions/641809/what-specifi...) question provide some interesting starting-points.