A class of simple matrix algebras

Fix an infinite set I and consider the associative matrix algebra MI(F) where F is a base field with char(F)≠2. For any couple of bijective maps σ,ν:I→I, such that σν=νσ and σ2=ν2, we introduce a linear subspace Ω(σ,ν) of MI(F). We endow it with a structure of (non-associative) algebra for a certain bilinear product, and obtain a wide class of non-associative algebras containing, in particular, the Lie algebras Lie(MI(F),t). We show that each algebra Ω(σ,ν) is simple.