Is there any one dimensional subalgebra which is an Ideal of the two dimensional non-abelian Lie Algebra?
i.e. is it invariant as a subalgebra of the 2D non-abelian algebra
I read that "all the subalgebras of an abelian algebra are automatically invariant"~Mathematical Methods for Physicists - George B. Arfken
I would assume this includes the trivial case of the group itself.
However, it then occurred to me that this would only hold if I consider the algebra as a subalgebra of itself and not necessarily in its own right.
Furthermore, can you prove with with the structure constants, generators or other methods of Lie Algebras that this is the case.
What motivated this is a tutorial in which, I proved there are no 2 dimensional semi-simple Lie Algebras by showing the metric was not invertible.
Another definition of semisimple is that it cannot contain any invariant abelian subalgebras. The abelian 2D algebra is clearly not semisimple. However it becomes a bit less trivial with the non-abelian 2D algebra. Clearly the only non trivial subalgebra of the non abelian 2D is the 1D subalgebra. The 1D algebra ticks the box for being abelian but can I also show it is an invariant subalgebra of the non-abelian 2D Lie algebra.
If so I can then get peace of mind and show there are no semi-simple non-abelian 2D lie Algebras!