The method of integration that I like most is via the theory of central extensions. To start with note that if $\mathfrak{g}$ is a subalgebra of $\mathfrak{gl}_n$, then the subgroup $\langle\exp(\mathfrak{g})\rangle$ of $GL_n$ that is generated by $\exp(\mathfrak{g})$ integrates $\mathfrak{g}$. Thus we are done with integrating $\mathfrak{g}$ if we find a faithful representation of $\mathfrak{g}$ (which exists by Ado's Theorem).

The problem with the above procedure is that the faithful representation of $\mathfrak{g}$ is not very canonical. However, there is a very canonical representation, the adjoint representation $\operatorname{ad}\colon\mathfrak{g}\to \mathfrak{der(\mathfrak{g}})\leq \mathfrak{gl}(\mathfrak{g})$. This is not faithful, the kernel is the center $\mathfrak{z}(\mathfrak{g})$ of $\mathfrak{g}$ and thus we have a canonical central extension $$\mathfrak{z}(\mathfrak{g})\to\mathfrak{g} \to \mathfrak{g}_{\operatorname{ad}}:=\operatorname{ad}(\mathfrak{g}). $$ We now set out to integrate this central extension. If we set $G_{\operatorname{ad}}:=\langle\exp(\mathfrak{g_{\operatorname{ad}}})\rangle\leq\operatorname{Aut}(\mathfrak{g})$, then this is a connected Lie group integrating $\mathfrak{g}_{\operatorname{ad}}$, and so is its universal covering $\widetilde{G}_{\operatorname{ad}}$. Since $\pi_2(\widetilde{G}_{\operatorname{ad}})=\pi_1(\widetilde{G}_{\operatorname{ad}})=0$ (this is the case for any simply connected finite-dimensional Lie group and was used in different disguise by Cartan in one of his first proofs of Lie's Third Theorem) there now exists a unique central extension $$\mathfrak{z}(\mathfrak{g})\to \widehat{G}\to \widetilde{G}_{\operatorname{ad}}$$ that integrates the above central extension of Lie algebras. The latter follows from the exact sequence$$\operatorname{Hom}(\pi_1(K),Z)\to \operatorname{Ext}(K,Z)\xrightarrow{L}\operatorname{Ext}(\mathfrak{k},\mathfrak{z})\to \operatorname{Hom}(\pi_2(K),Z)\oplus\operatorname{Hom}(\pi_1(K),...) $$ (where $L$ is the Lie differentiation and $...$ is some abelian group not of big importance for this discussion) that can be found in the paper of Neeb that was already cited by Peter Michor (and also in Theorem 7.12 of Neeb's "Central extensions of infinite-dimensional Lie groups" Ann. Inst. Fourier (Grenoble) 52 (2002)(5):1365–1442 (MR1935553)).

The nice thing about the latter approach is that

• it is functorial at each step
• it also works in infinite dimensions
• can (in infinite-dimensions) be used to show that certain Lie algebras do not integrate
• can be used in the non-integrable case to show how non-integrable Lie algebras still integrate to etale Lie 2-groups

I could elaborate on any of the above points, but this is perhaps a bit off topic from the original question.

The method of integration that I like most is via the theory of central extensions. To start with note that if $\mathfrak{g}$ is a subalgebra of $\mathfrak{gl}_n$, then the subgroup $\langle\exp(\mathfrak{g})\rangle$ of $GL_n$ that is generated by $\exp(\mathfrak{g})$ integrates $\mathfrak{g}$. Thus we are done with integrating $\mathfrak{g}$ if we find a faithful representation of $\mathfrak{g}$ (which exists by Ado's Theorem).

The problem with the above procedure is that the faithful representation of $\mathfrak{g}$ is not very canonical. However, there is a very canonical representation, the adjoint representation $\operatorname{ad}\colon\mathfrak{g}\to \mathfrak{der(\mathfrak{g}})\leq \mathfrak{gl}(\mathfrak{g})$. This is not faithful, the kernel is the center $\mathfrak{z}(\mathfrak{g})$ of $\mathfrak{g}$ and thus we have a canonical central extension $$\mathfrak{z}(\mathfrak{g})\to\mathfrak{g} \to \mathfrak{g}_{\operatorname{ad}}:=\operatorname{ad}(\mathfrak{g}). $$ We now set out to integrate this central extension. If we set $G_{\operatorname{ad}}:=\langle\exp(\mathfrak{g_{\operatorname{ad}}})\rangle\leq\operatorname{Aut}(\mathfrak{g})$, then this is a connected Lie group integrating $\mathfrak{g}_{\operatorname{ad}}$, and so is its universal covering $\widetilde{G}_{\operatorname{ad}}$. Since $\pi_2(\widetilde{G}_{\operatorname{ad}})=\pi_1(\widetilde{G}_{\operatorname{ad}})=0$ (this is the case for any simply connected finite-dimensional Lie group and was used in different disguise by Cartan in one of his first proofs of Lie's Third Theorem) there now exists a unique central extension $$\mathfrak{z}(\mathfrak{g})\to \widehat{G}\to \widetilde{G}_{\operatorname{ad}}$$ that integrates the above central extension of Lie algebras. The latter follows from the exact sequence$$\operatorname{Hom}(\pi_1(K),Z)\to \operatorname{Ext}(K,Z)\xrightarrow{L}\operatorname{Ext}(\mathfrak{k},\mathfrak{z})\to \operatorname{Hom}(\pi_2(K),Z)\oplus\operatorname{Hom}(\pi_1(K),...) $$ (where $L$ is the Lie differentiation and $...$ is some abelian group not of big importance for this discussion) that can be found in the paper of Neeb that was already cited by Peter Michor (and also in Theorem 7.12 of Neeb's "Central extensions of infinite-dimensional Lie groups" Ann. Inst. Fourier (Grenoble) 52 (2002)(5):1365–1442 (MR1935553)).

The nice thing about the latter approach is that

• it is functorial at each step
• it also works in infinite dimensions
• it can (in infinite-dimensions) be used to show that certain Lie algebras do not integrate
• it can be used in the non-integrable case to show how non-integrable Lie algebras still integrate to etale Lie 2-groups

I could elaborate on any of the above points, but this is perhaps a bit off topic from the original question.

The method of integration that I like most is via the theory of central extensions. To start with note that if $\mathfrak{g}$ is a subalgebra of $\mathfrak{gl}_n$, then the subgroup $\langle\exp(\mathfrak{g})\rangle$ of $GL_n$ that is generated by $\exp(\mathfrak{g})$ integrates $\mathfrak{g}$. Thus we are done with integrating $\mathfrak{g}$ if we find a faithful representation of $\mathfrak{g}$ (which exists by Ado's Theorem).

The problem with the above procedure is that the faithful representation of $\mathfrak{g}$ is not very canonical. However, there is a very canonical representation, the adjoint representation $\operatorname{ad}\colon\mathfrak{g}\to \mathfrak{der(\mathfrak{g}})\leq \mathfrak{gl}(\mathfrak{g})$. This is not faithful, the kernel is the center $\mathfrak{z}(\mathfrak{g})$ of $\mathfrak{g}$ and thus we have a canonical central extension $$\mathfrak{z}(\mathfrak{g})\to\mathfrak{g} \to \mathfrak{g}_{\operatorname{ad}}:=\operatorname{ad}(\mathfrak{g}). $$ We now set out to integrate this central extension. If we set $G_{\operatorname{ad}}:=\langle\exp(\mathfrak{g_{\operatorname{ad}}})\rangle\leq\operatorname{Aut}(\mathfrak{g})$, then this is a connected Lie group integrating $\mathfrak{g}_{\operatorname{ad}}$, and so is its universal covering $\widetilde{G}_{\operatorname{ad}}$. Since $\pi_2(\widetilde{G}_{\operatorname{ad}})=\pi_1(\widetilde{G}_{\operatorname{ad}})=0$ there now exists a unique central extension $$\mathfrak{z}(\mathfrak{g})\to \widehat{G}\to \widetilde{G}_{\operatorname{ad}}$$ that integrates the above central extension of Lie algebras. The latter follows from the exact sequence$$\operatorname{Hom}(\pi_1(K),Z)\to \operatorname{Ext}(K,Z)\xrightarrow{L}\operatorname{Ext}(\mathfrak{k},\mathfrak{z})\to \operatorname{Hom}(\pi_2(K),Z)\oplus\operatorname{Hom}(\pi_1(K),...) $$ (where $L$ is the Lie differentiation and $...$ is some abelian group not of big importance for this discussion) that can be found in the paper of Neeb that was already cited by Peter Michor (and also in Theorem 7.12 of Neeb's "Central extensions of infinite-dimensional Lie groups" Ann. Inst. Fourier (Grenoble) 52 (2002)(5):1365–1442 (MR1935553)).

The nice thing about the latter approach is that

• it is functorial at each step
• it also works in infinite dimensions
• can (in infinite-dimensions) be used to show that certain Lie algebras do not integrate
• can be used in the non-integrable case to show how non-integrable Lie algebras still integrate to etale Lie 2-groups

I could elaborate on any of the above points, but this is perhaps a bit off topic from the original question.

The method of integration that I like most is via the theory of central extensions. To start with note that if $\mathfrak{g}$ is a subalgebra of $\mathfrak{gl}_n$, then the subgroup $\langle\exp(\mathfrak{g})\rangle$ of $GL_n$ that is generated by $\exp(\mathfrak{g})$ integrates $\mathfrak{g}$. Thus we are done with integrating $\mathfrak{g}$ if we find a faithful representation of $\mathfrak{g}$ (which exists by Ado's Theorem).

The problem with the above procedure is that the faithful representation of $\mathfrak{g}$ is not very canonical. However, there is a very canonical representation, the adjoint representation $\operatorname{ad}\colon\mathfrak{g}\to \mathfrak{der(\mathfrak{g}})\leq \mathfrak{gl}(\mathfrak{g})$. This is not faithful, the kernel is the center $\mathfrak{z}(\mathfrak{g})$ of $\mathfrak{g}$ and thus we have a canonical central extension $$\mathfrak{z}(\mathfrak{g})\to\mathfrak{g} \to \mathfrak{g}_{\operatorname{ad}}:=\operatorname{ad}(\mathfrak{g}). $$ We now set out to integrate this central extension. If we set $G_{\operatorname{ad}}:=\langle\exp(\mathfrak{g_{\operatorname{ad}}})\rangle\leq\operatorname{Aut}(\mathfrak{g})$, then this is a connected Lie group integrating $\mathfrak{g}_{\operatorname{ad}}$, and so is its universal covering $\widetilde{G}_{\operatorname{ad}}$. Since $\pi_2(\widetilde{G}_{\operatorname{ad}})=\pi_1(\widetilde{G}_{\operatorname{ad}})=0$ (this is the case for any simply connected finite-dimensional Lie group and was used in different disguise by Cartan in one of his first proofs of Lie's Third Theorem) there now exists a unique central extension $$\mathfrak{z}(\mathfrak{g})\to \widehat{G}\to \widetilde{G}_{\operatorname{ad}}$$ that integrates the above central extension of Lie algebras. The latter follows from the exact sequence$$\operatorname{Hom}(\pi_1(K),Z)\to \operatorname{Ext}(K,Z)\xrightarrow{L}\operatorname{Ext}(\mathfrak{k},\mathfrak{z})\to \operatorname{Hom}(\pi_2(K),Z)\oplus\operatorname{Hom}(\pi_1(K),...) $$ (where $L$ is the Lie differentiation and $...$ is some abelian group not of big importance for this discussion) that can be found in the paper of Neeb that was already cited by Peter Michor (and also in Theorem 7.12 of Neeb's "Central extensions of infinite-dimensional Lie groups" Ann. Inst. Fourier (Grenoble) 52 (2002)(5):1365–1442 (MR1935553)).

The nice thing about the latter approach is that

• it is functorial at each step
• it also works in infinite dimensions
• can (in infinite-dimensions) be used to show that certain Lie algebras do not integrate
• can be used in the non-integrable case to show how non-integrable Lie algebras still integrate to etale Lie 2-groups

I could elaborate on any of the above points, but this is perhaps a bit off topic from the original question.