[proofplan] Choose ordered bases of $\mathfrak n^-$, $\mathfrak h$, and $\mathfrak n^+$, and concatenate them in triangular order to obtain an ordered basis of $\mathfrak g$. The Poincare-Birkhoff-Witt theorem gives a basis of $U(\mathfrak g)$ consisting of ordered monomials in this concatenated basis, and those monomials are exactly products with all $\mathfrak n^-$ factors first, then all $\mathfrak h$ factors, then all $\mathfrak n^+$ factors. Applying PBW separately to the three Lie subalgebras identifies the three groups of ordered monomials with bases of their enveloping algebras. The multiplication map sends the [tensor product](/page/Tensor%20Product) of these three bases bijectively onto the triangular PBW basis of $U(\mathfrak g)$, hence is a [vector space](/page/Vector%20Space) isomorphism. [/proofplan] [step:Choose an ordered basis compatible with the triangular decomposition] Let \begin{align*} (x_1,\dots,x_a) \end{align*} be an ordered complex basis of $\mathfrak n^-$, let \begin{align*} (y_1,\dots,y_b) \end{align*} be an ordered complex basis of $\mathfrak h$, and let \begin{align*} (z_1,\dots,z_c) \end{align*} be an ordered complex basis of $\mathfrak n^+$. Since \begin{align*} \mathfrak g=\mathfrak n^- \oplus \mathfrak h \oplus \mathfrak n^+ \end{align*} is a [direct sum](/page/Direct%20Sum) of complex vector spaces, the concatenated ordered list \begin{align*} (x_1,\dots,x_a,y_1,\dots,y_b,z_1,\dots,z_c) \end{align*} is an ordered complex basis of $\mathfrak g$. [/step] [step:Identify the triangular PBW basis of $U(\mathfrak g)$] Let $i_{\mathfrak g}:\mathfrak g\to U(\mathfrak g)$ denote the canonical [Lie algebra](/page/Lie%20Algebra) map into the universal enveloping algebra. By [citetheorem:8827] applied to the ordered basis chosen above, the monomials \begin{align*} i_{\mathfrak g}(x_1)^{\alpha_1}\cdots i_{\mathfrak g}(x_a)^{\alpha_a}i_{\mathfrak g}(y_1)^{\beta_1}\cdots i_{\mathfrak g}(y_b)^{\beta_b}i_{\mathfrak g}(z_1)^{\gamma_1}\cdots i_{\mathfrak g}(z_c)^{\gamma_c} \end{align*} with $\alpha_i,\beta_j,\gamma_k\in\mathbb Z_{\ge 0}$ form a complex basis of $U(\mathfrak g)$. Denote this basis by $\mathcal B_{\mathfrak g}$. The hypotheses of PBW are satisfied because $\mathfrak g$ is a Lie algebra over $\mathbb C$ and the displayed concatenated list is an ordered basis of $\mathfrak g$. [/step] [step:Apply PBW separately to the three triangular subalgebras] Let $i_-:\mathfrak n^-\to U(\mathfrak n^-)$, $i_0:\mathfrak h\to U(\mathfrak h)$, and $i_+:\mathfrak n^+\to U(\mathfrak n^+)$ be the canonical Lie algebra maps. Applying [citetheorem:8827] to the ordered bases of $\mathfrak n^-$, $\mathfrak h$, and $\mathfrak n^+$ gives complex bases \begin{align*} \mathcal B_-=\{i_-(x_1)^{\alpha_1}\cdots i_-(x_a)^{\alpha_a}:\alpha_i\in\mathbb Z_{\ge 0}\} \end{align*} of $U(\mathfrak n^-)$, \begin{align*} \mathcal B_0=\{i_0(y_1)^{\beta_1}\cdots i_0(y_b)^{\beta_b}:\beta_j\in\mathbb Z_{\ge 0}\} \end{align*} of $U(\mathfrak h)$, and \begin{align*} \mathcal B_+=\{i_+(z_1)^{\gamma_1}\cdots i_+(z_c)^{\gamma_c}:\gamma_k\in\mathbb Z_{\ge 0}\} \end{align*} of $U(\mathfrak n^+)$. Again, the PBW hypotheses are satisfied because each of $\mathfrak n^-$, $\mathfrak h$, and $\mathfrak n^+$ is a Lie algebra over $\mathbb C$ equipped with the indicated ordered basis. [guided] We now apply the same theorem three more times, once to each piece of the triangular decomposition. For $\mathfrak n^-$, the ordered basis is $(x_1,\dots,x_a)$, so [citetheorem:8827] says that the ordered monomials \begin{align*} i_-(x_1)^{\alpha_1}\cdots i_-(x_a)^{\alpha_a} \end{align*} with $\alpha_i\in\mathbb Z_{\ge 0}$ form a complex basis of $U(\mathfrak n^-)$. For $\mathfrak h$, the ordered basis is $(y_1,\dots,y_b)$, so the ordered monomials \begin{align*} i_0(y_1)^{\beta_1}\cdots i_0(y_b)^{\beta_b} \end{align*} with $\beta_j\in\mathbb Z_{\ge 0}$ form a complex basis of $U(\mathfrak h)$. For $\mathfrak n^+$, the ordered basis is $(z_1,\dots,z_c)$, so the ordered monomials \begin{align*} i_+(z_1)^{\gamma_1}\cdots i_+(z_c)^{\gamma_c} \end{align*} with $\gamma_k\in\mathbb Z_{\ge 0}$ form a complex basis of $U(\mathfrak n^+)$. The reason for doing this separately is that the tensor product basis of \begin{align*} U(\mathfrak n^-)\otimes_{\mathbb C}U(\mathfrak h)\otimes_{\mathbb C}U(\mathfrak n^+) \end{align*} is built from one PBW monomial in each factor. We will compare those tensor product basis elements with the single PBW basis of $U(\mathfrak g)$ obtained from the concatenated triangular basis. [/guided] [/step] [step:Show that multiplication sends the tensor product basis bijectively to the triangular PBW basis] The inclusions $\mathfrak n^-\subset\mathfrak g$, $\mathfrak h\subset\mathfrak g$, and $\mathfrak n^+\subset\mathfrak g$ induce algebra homomorphisms $j_-$, $j_0$, and $j_+$ satisfying \begin{align*} j_-(i_-(x_i))=i_{\mathfrak g}(x_i), \qquad j_0(i_0(y_j))=i_{\mathfrak g}(y_j), \qquad j_+(i_+(z_k))=i_{\mathfrak g}(z_k). \end{align*} Therefore, for every choice of exponents $\alpha_i,\beta_j,\gamma_k\in\mathbb Z_{\ge 0}$, the multiplication map sends \begin{align*} i_-(x_1)^{\alpha_1}\cdots i_-(x_a)^{\alpha_a}\otimes i_0(y_1)^{\beta_1}\cdots i_0(y_b)^{\beta_b}\otimes i_+(z_1)^{\gamma_1}\cdots i_+(z_c)^{\gamma_c} \end{align*} to \begin{align*} i_{\mathfrak g}(x_1)^{\alpha_1}\cdots i_{\mathfrak g}(x_a)^{\alpha_a}i_{\mathfrak g}(y_1)^{\beta_1}\cdots i_{\mathfrak g}(y_b)^{\beta_b}i_{\mathfrak g}(z_1)^{\gamma_1}\cdots i_{\mathfrak g}(z_c)^{\gamma_c}. \end{align*} This assignment is bijective from the tensor product basis \begin{align*} \mathcal B_-\otimes\mathcal B_0\otimes\mathcal B_+ \end{align*} onto $\mathcal B_{\mathfrak g}$, because both basis elements are indexed by the same tuples of nonnegative integers \begin{align*} (\alpha_1,\dots,\alpha_a,\beta_1,\dots,\beta_b,\gamma_1,\dots,\gamma_c). \end{align*} [/step] [step:Conclude that the multiplication map is an isomorphism and expansions are unique] A complex-[linear map](/page/Linear%20Map) that sends a basis bijectively onto a basis is a vector space isomorphism. Since $m$ sends the basis \begin{align*} \mathcal B_-\otimes\mathcal B_0\otimes\mathcal B_+ \end{align*} bijectively onto $\mathcal B_{\mathfrak g}$, the map \begin{align*} m:U(\mathfrak n^-)\otimes_{\mathbb C}U(\mathfrak h)\otimes_{\mathbb C}U(\mathfrak n^+)\to U(\mathfrak g) \end{align*} is a vector space isomorphism. Because $\mathcal B_{\mathfrak g}$ is a basis of $U(\mathfrak g)$, every element of $U(\mathfrak g)$ has a unique finite complex linear expansion in the triangular PBW monomials. Equivalently, after identifying those monomials with the images under $m$ of the tensor product PBW basis, every element has a unique finite expansion as a linear combination of products \begin{align*} j_-(u)j_0(v)j_+(w) \end{align*} with $u\in\mathcal B_-$, $v\in\mathcal B_0$, and $w\in\mathcal B_+$. This is exactly the claimed triangular PBW factorisation. [/step]