Palindromes in the free metabelian Lie algebras

A palindrome, in general, is a word in a fixed alphabet which is preserved when taken in reverse order. Let F-2 be the free metabelian Lie algebra over a field of characteristic zero generated by x(1), x(2). We propose the following definition of palindromes in the setting of Lie algebras: An element f(x(1), x(2)) is an element of F-2 is called a palindrome if it is preserved under the change of generators; i.e. f(x(1), x(2)) = f(x(2), x(1)). We give a linear basis and an explicit infinite generating set for the Lie subalgebra of palindromes.