I am taking a Lie algebras course as a prerequisite to study Lie groups. The idea of a radical of a Lie algebra (maximal solvable ideal) has been defined in class but no other statements or theorems are available.
I have a question that asks me to calculate the radical of $\mathfrak{gl}(2,\mathbb{C})$, which is the general linear Lie algebra of $2 \times 2$ matrices with complex entries.
I just have no idea how to start the question; I know how to check something is an ideal of $\mathfrak{gl}(2,\mathbb{C})$ but I have no idea how to check it's a maximal ideal, or how to check that it's maximal with respect to solvable Lie algebras.
I thought of just guessing a maximal ideal (from a list of common ideals, like $\mathfrak{u}(2,\mathbb{C})$ (upper triangular matrices)) and using the correspondence theorem to prove no bigger ideal exists, but I couldn't get anywhere with it.