In this paper we provide a general tool to prove the consistency of
I1(λ) with various combinatorial properties at λ typical at
settings with 2λ>λ+, 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 λ+ and λ++