Please read this post as a naive follow up on a previous question.
Let $X$ be a Banach space and let $(I(X),\alpha)$ denote its injective envelope (e.g., CohenLacey1969). A low hanging fruit is the fact that if $X\oplus I(X)$ is $\lambda$-extensible, then $X$ is $\lambda$-injective.
It is perhaps relevant for the question that $I(X)$ is a rigid extension of $X$; that is, if $T:I(X)\to I(X)$ is a contraction and $T\circ\alpha = \alpha$, then $T$ is an isometry.
Q: Is there a relationship between $1$-extensibility of $X$ and a subset of bounded linear operators on $I(X)$?
The concepts $\mathcal{L}^{\infty}$-envelopes, $\mathcal{C}$-extension property in the literature gives a clue about that this relationship was probably considered before. I'd be grateful if you could direct me to a related reference.
Thanks very much in advance.
