Domia is the Rider sword used by the Elven Rider Dair.
It was found as an unnamed sword in Galbatorix's possession. Many theorize it was the Rider sword that belonged to Shruikan's original Rider, though there is no proof. When it was given to Dair, he carved the glyph for "domia", which means dominance in the Ancient Language, into the blade and sheath using magic.
It has an intricately designed gold and black handle, with a spiral-like handle for better grip, and a similarly-designed sheath. The pommel contains a brilliant onyx which has energy stored in it by both Dair and his dragon Rorkin. Though they only recently obtained it, the amount of energy stored in it is quite large, enough to cast spells that would normally kill him.
The hand guard is made of gold and designed to look like a stylized dragon head and wings. The blade a very dark black and is rather short, only about two feet in length, but fits Dair's fighting style very well, and can be used as a one-handed and two-handed sword, though Dair tends to use only one hand so he could cast spells with the other. The glyph for "domia" is carved into the blade, though it's very difficult to see.
Much like Oromis once did, Dair bound his wards to his sword so that it would draw energy from the onyx embedded in the pommel rather than from him. This way, he could fight for a longer period of time compared to those whose wards sapped energy from themselves.
However, if the sword was lost, he would lose all of his wards, and a few backup ones for when the sword was gone would start, but would sap the energy from Dair instead, causing him to slowly lose stamina, strength, and speed.