The 1-qubit Bell states are given by
|+⟩=1√2(|0⟩+|1⟩)|−⟩=1√2(|0⟩−|1⟩)
These are ortho-normal eigen-states of X:
X|+⟩=+1|+⟩X|−⟩=−1|−⟩
Taking tensor products gives a 2-qubit ortho-normal basis:
|++⟩≡|+⟩⊗|+⟩=12(|00⟩+|01⟩+|10⟩+|11⟩)|+−⟩≡|+⟩⊗|−⟩=12(|00⟩−|01⟩+|10⟩−|11⟩)|−+⟩≡|−⟩⊗|+⟩=12(|00⟩+|01⟩−|10⟩−|11⟩)|−−⟩≡|−⟩⊗|−⟩=12(|00⟩−|01⟩−|10⟩+|11⟩)
Note that, by construction, the four vectors in this 2-qubit basis do not describe 2-body entanglement since they are all tensor products. Linear combinations do, giving the 2-qubit Bell states:
|Φ+⟩=1√2(|00⟩+|11⟩)=1√2(|++⟩+|−−⟩)|Φ−⟩=1√2(|00⟩−|11⟩)=1√2(|−+⟩+|+−⟩)|Ψ+⟩=1√2(|01⟩+|10⟩)=1√2(|++⟩−|−−⟩)|Ψ−⟩=1√2(|01⟩−|10⟩)=1√2(|−+⟩−|+−⟩)
These states cannot be factorized as tensor products.
Another 1-qubit basis is the two eigen-states of Y:
|∘⟩=1√2(|0⟩+i|1⟩)|⋆⟩=1√2(|0⟩−i|1⟩)
Taking tensor products gives another 2-qubit ortho-normal basis:
|∘∘⟩≡|∘⟩⊗|∘⟩=12(|00⟩+i|01⟩+i|10⟩−|11⟩)|∘⋆⟩≡|∘⟩⊗|⋆⟩=12(|00⟩−i|01⟩+i|10⟩+|11⟩)|⋆∘⟩≡|⋆⟩⊗|∘⟩=12(|00⟩+i|01⟩−i|10⟩+|11⟩)|⋆⋆⟩≡|⋆⟩⊗|⋆⟩=12(|00⟩−i|01⟩−i|10⟩−|11⟩)
I kind of like this notation: the star suggests complex conjugation.