\overline{a * b} &=& \rm{Re}(a) \rm{Re}(b) - \rm{Im}(a)\rm{Im}(b) - (\rm{Re}(a) \rm{Im}(b)   \rm{Re}(b) \rm{Im}(a)) \; i \\
&=& \overline{a} * \overline{b}
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