What is a classification of all algebras $A$ (purely algebraic algebras, Banach or $C^*$ algebras or Lie algebras) with the following property:

Every finite codimensional subalgebra $B$ of $A$ contains a subalgebra $I$ such that $I$ is an ideal in $A$ and $A/I$ is a finite dimensional algebra.

One can ask a similar question for "Rings" but consider finite quotient rather than finite codimensionality in algebra case.

This question is inspired by this post and its comments conversation