Most general definition of Borel and parabolic Lie algebras?

There are certainly sources defining Borel subalgebras and parabolic subalgebras in general, e.g., in the book "Lie Algebras and Algebraic Groups" by Patrice Tavel and Rupert W. T. Yu. "We generalise the definitions of Borel subalgebras and parabolic subalgebras to arbitrary Lie algebras and establish the relations between the group objects and the Lie algebra objects".

This contains the details we want (Chapter $29$).

At least for $\mathfrak{g}$ real semisimple there is a nice definition without using maximal solvable Lie subalgebras. A subalgebra $\mathfrak{p}\leq\mathfrak{g}$ is parabolic if $\mathfrak{p}^\perp\leq\mathfrak{p}$ is nilpotent, where $\mathfrak{p}^\perp$ is the polar with respect to the Killing form. I think this definition could be extended to other cases as well.