I was looking for an article mimicing result of Pelczynski for $\ell_p$. I have found this one
Rodriguez-Salinas, B. (1994). On the Complemented Subspaces of $c_0(I)$ and $\ell_p(I)$ for $1 < p < \infty$, Atti Sem. Mat. Fis. Univ. Modena, 42, 399–402.
but I can't download it anywhere. Could someone provide me with this article or show main steps in that proofs.
This is just an exercise in the reflexive range once you know the proof in the separable case. Suppose $X$ is a subspace of $\ell_p(I)$ with density character $\aleph$. Since each vector in $X$ has countable support, WLOG $I=\aleph$. If $X$ is complemented, you just need to check that $X$ has a complemented subspace isomorphic to $\ell_p(\aleph)$, for then you can apply the decomposition method. The separable case is known, so assume that $\aleph$ is uncountable. Take a maximal set of disjointly supported unit vectors in $X$. If the cardinality of the set is less than $\aleph$, then the union $A$ of their supports has cardinality less than $\aleph$. Let $P$ be the band projection onto the functions supported on $A$. Since $X$ has density character greater than $|A|$, this projection is not one to one (the adjoint cannot have dense range; this is where reflexivity helps).
