I am using the notation that $g$ is the Lie algebra of the Lie group $G$ and $T$ is the maximal torus of $G$ and $t$ is the Lie algebra of $T$ (and hence $t$ is the Cartan subalgebra of $g$). A superscript of $^*$ would indicate a dual vector space.
From the literature I seem to get two "different" ways of thinking about roots of a Lie Algebra,
They are the non-trivial weights of the adjoint representation of $G$ on $g$ i.e they are the non-trivial representations of $T$ that occur when the Adjoint representation of $G$ on $g$ is restricted to $T$.
They are the elements in $t^*$ which need to act on the elements of $t$ to give the eigen-values of the commutator/adjoint action of the elements of $t$ (complexified) on $g/t$ (complexified).
Some aspects of this which are not clear to me are,
Its not clear that the above two pictures are "equal" but I guess one can map from one to the other. But is there an explicit bijection between the two descriptions above?
Given the structure of representations of a torus I guess here it follows that in the second picture the elements of $t^*$ that are talked of are precisely those that map the $\mathbb{Z}$-lattice of $t$ to $\mathbb{Z}$. But I would like to know of a clear proof.
To do the second picture was it necessary to complexify $t$ and $\cal{g}/t$?
When one has complexified the vector spaces then how does one do the thinking in the language of $\mathbb{Z}$-lattice of $t$ being mapped to $\mathbb{Z}$? Is there a canonical way to embed the lattice?