Dimensions of exceptional simple Lie algebras

Given: The explicit reduced crystallographic root systems of types $G_2,F_4,E_6,E_7,E_8$ described in the cited source.

AC is assumed and is used through the existence and classification theorems (The Axiom of Choice).

In the explicit models of the cited source, $G_2$ has the twelve roots listed in Example 21.9 and rank $2$; Definitions 23.8, 23.11, 23.14 and 23.15 give respectively $48,240,126,72$ roots for $F_4,E_8,E_7,E_6$, whose ranks are respectively $4,8,7,6$.

For a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$ and root system $\mathrm{\Phi }$ one has $\dim \mathfrak{g}=\dim \mathfrak{h}+\mathrm{\mid }\mathrm{\Phi }\mathrm{\mid }$. Moreover the real root span is identified with the real dual of a real form of $\mathfrak{h}$, so ${\dim }_{\mathbb{C}}\mathfrak{h}={\dim }_{\mathbb{R}}{\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }=\mathrm{rank}\mathrm{\Phi }$ (Dimension formula from roots, Roots of a complex semisimple Lie algebra form a reduced crystallographic root system, Rank and isomorphism of root systems).

Every reduced crystallographic root system is the root system of a finite-dimensional complex simple Lie algebra when irreducible, and the type determines the isomorphism class (Existence theorem for complex semisimple Lie algebras, Cartan-Killing classification of complex simple Lie algebras).