We are looking for the maximum volume of an axis-parallel cuboid that can be inscribed into a given paralleloid. We're looking to maximize $$V:\mathbb{R_+^3\rightarrow R}, V(x,y,z)=2x2y2z, \text{under the constraint that,}\ g(x,y,z)=\bigg(\frac{x}{a}\bigg)^2+\bigg(\frac{y}{b}\bigg)^2+\bigg(\frac{z}{c}\bigg)^2-1=0$$
I'm somewhat familiar with the lagrange multiplier used for maximization problems but I could't manage much in this problem. Utilizing the lagrange multiplier, we wish to solve the system of $$\nabla f(x,y,z)=\lambda\nabla g(x,y,z) \\ \implies 8yz=2\lambda\frac{x}{a^2} ,\ 8xz=2\lambda \frac{y}{b^2},\ 8xy=2\lambda\frac{z}{c^2} $$ I know the expression can be simplified via algebra but I couldn't find a solution.