In chapter 19 of The Lightning Thief, how do Annabeth and Grover display loyalty toward Percy?

In chapter nineteen, Annabeth and Grover follow Percy to Elysium, almost get sucked into Tartarus, and finally enter the palace of Hades together. But in order to leave the Underworld, they need to use the pearls—and they only have three of them. Percy's mom is there, which means they would need four pearls in order for all four of them to leave. Hades explains Percy's predicament:

"What a shame. You do realize each only protects a single person. Try to take your mother, then, little godling. And which of your friends will you leave behind to spend eternity with me? Go on. Choose. Or give me the backpack and accept my terms."

Annabeth and Grover are not afraid. They prove their loyalty to their friend by volunteering to remain in the Underworld so Percy can save his mom. First Grover offers to stay behind:

"I'm a satyr," Grover said. "We don't have souls like humans do. He can torture me until I die, but he won't get me forever. I'll just be reincarnated as a flower or something. It's the best way."

Annabeth explains that Percy will need Grover to protect him, so she should be the one to stay. She is willing to die for her friends:

"Get his mom out of here. I'll cover you. I plan to go down fighting."

Annabeth and Grover both show extreme bravery and loyalty. Percy reflects on how they have been loyal to him this whole time.

"Stop it, both of you!" I felt like my heart was being ripped in two. They had both been with me through so much. I remembered Grover dive-bombing Medusa in the statue garden, and Annabeth saving us from Cerberus; we'd survived Hephaestus's Waterland ride, the St. Louis Arch, the Lotus Casino. I had spent thousands of miles worried that I'd be betrayed by a friend, but these friends would never do that. They had done nothing but save me, over and over, and now they wanted to sacrifice their lives for my mom.