Can your commander be your general if it is a background and pairs with another card?

In Commander/EDH, any legendary card that is legal for the format can serve as your commander. Backgrounds are a special subtype of enchantment creatures introduced in Commander Legends: Battle for Baldur's Gate. Because they are legendary enchantment creatures, they are perfectly legal as commanders under CR 903.3.

The 'Choose a Background' mechanic (CR 702.124) allows a legendary creature with that ability to have a Background as a second commander alongside it. Both cards together form your commander pair — this is the intended use case. Your Background commander grants its benefits when you control a commander with 'Choose a Background.'

You can run a Background as your sole commander, but its passive ability that says 'as long as you control your commander' would never trigger its bonus text, since your commander (the Background itself) rarely has the 'Choose a Background' ability. The Background is still a legal commander, you just lose access to the synergistic bonus.

Example: You choose Haunted One (a Background) as your sole commander. It enters the battlefield as a 0/0 enchantment creature, but its bonus ability — granting your commander menace and a triggered ability — never activates because Haunted One itself is not a 'Choose a Background' creature. Contrast this with pairing it alongside Burakos, Party Leader, where Burakos gains all those bonuses.

So the short answer: yes, a Background is a valid commander, but its design intent is to be the second commander paired with a 'Choose a Background' legendary creature. Running it alone is legal but largely ineffective for its bonus text.