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$\begingroup$Hint: use the polarization identity to show the relation for any $a,b$ instead of just the quadratic form with $a,a$. See this for instance math.stackexchange.com/a/3035810/399263$\endgroup$
$\begingroup$Once you showed by polarization that this implies $(T(a)|b)=(T^*(a)|b)$ for all $a,b$ this implies $((T-T^*)(a),b)=0$ for all $a,b$ thus for $b=(T-T^*)(a)$ you get $((T-T^*)(a)|(T-T^ *)(a))=0$ for all $a$, thus $(T-T^*)(a)=0$ for all $a$, thus $T=T^*$$\endgroup$