Corollary

Let $\mathfrak{t}$ and $\mathfrak{t}'$ be Cartan subalgebras of $\mathfrak{g},\mathfrak{a}$ a subset of $\mathfrak{t}$, and $u$ an automorphism of $\mathfrak{g}$ that takes $\mathfrak{a}$ into $\mathfrak{t}'$. There exists an element $v$ of Int($\mathfrak{g}$) such that $u\circ v$ takes $\mathfrak{t}$ to $\mathfrak{t}'$, and coincides with $u$ on $\mathfrak{a}$.

Put G = Int($\mathfrak{g}$), and consider the fixer $Z_G(\mathfrak{a})$ of $\mathfrak{a}$ in G; this is a Lie subgroup of G, whose Lie algebra $\mathfrak{z}_{\mathfrak{g}}(\mathfrak{a})$ consists of the elements of $\mathfrak{g}$ that commute with every element of $\mathfrak{a}$ (Chap. III, §9, no. 3, Prop. 7). Then $\mathfrak{t}$ and $u^{-1}(\mathfrak{t}')$ are two Cartan subalgebras of the compact Lie algebra $\mathfrak{z}_{\mathfrak{g}}(\mathfrak{a})$. By Th. 1, there exists an element $v$ of $Z_G(\mathfrak{a})$ such that $v(\mathfrak{t}) =u^{-1}(\mathfrak{t}')$; any such element has the desired properties.