In this question "constructing" and "doubling" is meant in the compass-and-straightedge sense.

On my desk I have five Basic Algebra texts treating constructability in the plane $\mathbb{C}$ or $\mathbb{R}^2$ as an application of basic field theory. After appropriate definitions of the possible construction steps, four of these, namely Hornfeck, Jacobson, Lorenzen, and Meyberg, prove that $\sqrt[3]{2}$ is inconstructible starting from $\{0,1\}$ or $\{(0,0),(1,0)\}$, respectively, but then conclude without further justification that the duplication of the cube is impossible.

For a while I believed this last step to be obvious. But now, having to teach this for the first time the day after tomorrow, I have doubts: Being given a cube, say in $\mathbb{R}^3$, should mean being given its eight corners, and then I could use these to do constructions in space, using lines through two points and circles around one point and through two points. Or, to put it differently, I could take any three noncollinear points given or already constructed and do plane constructions in the plane spanned by these. Restricting the constructions initially to one particular coordinate plane containing one face of the cube appears unjustified to me.

My specific questions are:

How does one treat this problem honestly and elegantly, with a minimum of coordinate computations?

Is the problem I see perhaps the reason why the fifth of my books, by M. Artin, does not mention cube doubling?