Complexification, complex structures, and linear ordinary differential equations

This is similar to how Cardano’s formula, which expresses the roots of a real cubic polynomial in terms of its coefficients, involves complex numbers and yet the final result may still be real. In the following, unless I specify the dimension of a vector space, any statement about real vector spaces is about real vector spaces of finite or infinite dimension, and any statement about complex vector spaces is about complex vector spaces of finite or infinite dimension.