I can summarize this opinion in three words:
"And" means and.
Under ICWA, an "Indian child" is "either (a) a member of an Indian tribe or (b) is eligible for membership in an Indian tribe and is the biological child of a member of an Indian tribe." It's undisputed that the child here is not a member of an Indian tribe, so (a) doesn't apply.
But he's also "eligible for membership" in the Cherokee tribe because his great-great grandfather was an enrolled member. So (b) might apply.
Except for the "and" part.
You've got to be eligible for membership and be the biological child of a member of the tribe. It's undisputed that the child isn't the biological child of a member of the tribe. His father isn't a member. His mother isn't a member.
"And" means and. You've got to both be eligible for membership and be the biological child of a member.
That's what the statute says.
End of story. As the Court of Appeal holds.