In this paper we provide a general tool to prove the consistency of
$I1(\lambda)$ with various combinatorial properties at $\lambda$ typical at
settings with $2^\lambda>\lambda^+$, that does not need a profound knowledge of
the forcing notions involved. Examples of such properties are the first failure
of GCH, a very good scale and the negation of the approachability property, or
the tree property at $\lambda^+$ and $\lambda^{++}$

Dimonte, Vincenzo