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We prove the $*$–version of the Joly–Becker theorem: a skew field admits a $*$–ordering of level $n$ iff it admits a $*$–ordering of level $n\ell$ for some (resp. all) odd $\ell \in \mathbb{N}$. For skew fields with an imaginary unit and fields stronger results are given: a skew field with imaginary unit that admits a $*$–ordering of higher level also admits a $*$–ordering of level $1$. Every field that admits a $*$–ordering of higher level admits a $*$–ordering of level $1$ or $2$