Standard and nonstandard extensions of Lie algebras

We study the problem of quadruple extensions of simple Lie algebras. We find that, adding a new simple root α+4⁠, it is not possible to have an extended Kac-Moody algebra described by a Dynkin-Kac diagram with simple links and no loops between the dots, while it is possible if α+4 is a Borcherds imaginary simple root. We also comment on the root lattices of these new algebras. The folding procedure is applied to the simply laced triple extended Lie algebras, obtaining all the nonsimply laced ones. Nonstandard extension procedures for a class of Lie algebras are proposed. It is shown that the two-extensions of E8⁠, with a dot simply linked to the Dynkin-Kac diagram of E9⁠, are rank 10 subalgebras of E10⁠. Finally the simple root systems of a set of rank 11 subalgebras of E11⁠, containing as sub-algebra E10⁠, are explicitly written.