Well since in the books elites are supposed to be the nightmare of humans...
Have the elite attempt an assassination, but the Spartan grabs it's arms. The elite then bites the spartans head and thrashes him (much like a dog does), breaking his spine. And have the Spartan screaming and clawing at the elite before abruptly getting silent and limp.
And as a Spartan on Spartan, a good old fashioned stab to the kidneys and a quick throat slice, dead in 2 seconds and not even able to let out a peep.
Or turn him around kick him in the crotch and follow up with the knife into his side.
Elite on elite, dual wrist blade into the back of the head and a kick to pull it out.
Spartan on elite, the ones they already have seem possible without becoming stronger than the elite so leave them as is.