A Banach space $E$ is injective when it is complemented in each Banach space $X$ that contains it as a closed subspace. The space $E$ is $1$-injective if the copy of $E$ in $X$ is the range of a norm-one projection. It has been known for more that 50 years that $E$ is $1$-injective if and only if it is linearly isometric to a Banach space of continuous functions $C(K)$ with $K$ a extremely disconnected compact. However it is not known if every injective space is isomorphic to a $1$-injective space.
I am interested in the variety of examples of $1$-injective spaces that are known, and in existing classifications of them. I know that there exists a injective space that it is not isomorphic to a dual space (Rosenthal), there exist injective spaces that are isomorphic to dual spaces, but not isomorphic to bidual spaces ($L_1(\mu)^*$ for $\mu$ a finite measure such that $L_1(\mu)$ is non-separable), and each injective space isomorphic to a bidual is isomorphic to $\ell_\infty(\Gamma)$ for some set $\Gamma$ (Haydon).
Are there good references for examples and/or classifications of injective Banach spaces?
Is every Banach space embeddable as a subspace in an injective space which is not isomorphic to a bidual space like $\ell_\infty(\Gamma)$ (not just $\ell_\infty(\Gamma)\oplus L_1(\mu)^*$)?
