Conjugacy of maximal tori

Choose the metric of [L1] and its inner product on $\mathfrak{g}$. Conjugation is a composite of left and right isometries fixing $e$, so its differential ${\mathrm{Ad}}_g$ preserves this inner product. Differentiate along $\exp (tX)$ using [L1] and [L3] to obtain $⟨[X,U],V⟩=-⟨U,[X,V]⟩$. Thus every ${\mathrm{ad}}_X$ is skew-adjoint.

If an abelian Lie subalgebra $\mathfrak{a}$ contains ${\mathfrak{t}}_i$, then $A={\exp }_G(\mathfrak{a})$ is a subgroup by [L3] and is abelian; it is connected as the continuous image of a vector space. Its closure $H$ is a connected compact subgroup by [L2]. To check the group and abelian assertions for the closure, continuity of multiplication, inversion and the commutator map extends the corresponding identities from the dense subset $A$ (and $A\times A$) to $H$ (and $H\times H$). The closed-subgroup theorem gives its embedded Lie structure. For $V\in \mathfrak{a}$, the curve ${\exp }_G(tV)$ lies in $H$, and submanifold charts make this ambient smooth curve smooth as an $H$-valued curve; hence its velocity $V$ lies in $\mathrm{Lie}H$. Naturality and local invertibility of the exponential on $T_i$ show that $A$ contains an identity neighborhood in $T_i$; the subgroup it generates is open and closed in connected $T_i$, hence is all of $T_i$. Thus $H$ is a torus containing $T_i$, so maximality gives $H=T_i$ and $\mathfrak{a}\subseteq {\mathfrak{t}}_i$. Therefore ${\mathfrak{t}}_i$ is maximal abelian.

Fix either $\mathfrak{t}={\mathfrak{t}}_i$. Extend the real inner product to the positive Hermitian product on ${\mathfrak{g}}_{\mathbb{C}}$ using a real orthonormal basis. The operators ${\mathrm{ad}}_H$, $H\in \mathfrak{t}$, remain skew-adjoint after complexification, so are normal and diagonalizable by [L4]. They commute because $[H,H^{\mathrm{\prime }}]=0$ and Jacobi gives $[{\mathrm{ad}}_H,{\mathrm{ad}}_{H^{\mathrm{\prime }}}]={\mathrm{ad}}_{[H,H^{\mathrm{\prime }}]}$. Simultaneous diagonalization yields finitely many joint eigenspaces with eigenvalue functions $\alpha :\mathfrak{t}\to \mathbb{C}$ that are real-linear, by linearity of $H\mapsto {\mathrm{ad}}_H$. The joint zero eigenspace is ${\mathfrak{t}}_{\mathbb{C}}$: if a real $U$ commutes with all of $\mathfrak{t}$, then $\mathfrak{t}+\mathbb{R}U$ is abelian, so $U\in \mathfrak{t}$ by step 1.2; for complex $U$ the real and imaginary parts separately commute.

For every nonzero eigenvalue function $\alpha$ in step 2.1, its real kernel is a proper subspace of $\mathfrak{t}$. By [L5] choose $X_i$ outside the finite union of these kernels. On each nonzero joint eigenspace ${\mathrm{ad}}_{X_i}$ has nonzero eigenvalue, and its kernel is therefore exactly ${\mathfrak{t}}_{i,\mathbb{C}}$. Intersecting with $\mathfrak{g}$ gives ${\mathfrak{c}}_{\mathfrak{g}}(X_i)={\mathfrak{t}}_i$. If the family of nonzero eigenvalue functions is empty, step 2.1 says ${\mathfrak{t}}_i=\mathfrak{g}$, and $X_i=0$ works, including the zero-dimensional case.

Put $X=X_1$ and $Y=X_2$. The smooth function $f(g)=⟨{\mathrm{Ad}}_{g}X,Y⟩$ attains a maximum at $g_0$ by compactness. For each $Z\in \mathfrak{g}$, differentiate $f(\exp (tZ)g_0)$ at zero. With $U={\mathrm{Ad}}_{g_0}X$, the derivative is $⟨[Z,U],Y⟩=⟨Z,[U,Y]⟩$ by step 1.1 and symmetry. Its vanishing for every $Z$ implies $[U,Y]=0$.

Step 3.1 gives $U\in \mathfrak{c}(Y)={\mathfrak{t}}_2$. Since ${\mathfrak{t}}_2$ is abelian, ${\mathfrak{t}}_2\subseteq \mathfrak{c}(U)={\mathrm{Ad}}_{g_0}\mathfrak{c}(X)={\mathrm{Ad}}_{g_0}{\mathfrak{t}}_1$. The last space is abelian because conjugation induces a Lie-algebra automorphism. Maximal abelianness of ${\mathfrak{t}}_2$ forces equality.

The connected embedded subgroups $g_0{T}_1{g}_0^{-1}$ and $T_2$ have the same Lie algebra by step 5.1, and therefore are the same subgroup by [L6]. This proves conjugacy. The argument includes trivial tori, the zero Lie algebra and the empty family of nonzero weights as treated in step 3.1. All choice requirements are covered by [A1]; no reductivity, Cartan-subalgebra recognition, root decomposition or torus lattice classification was invoked.