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There are some known, quite explicit, algorithms for the case in which $\mathfrak h$ is abelian. The fact that this case is already quite difficult shows how the problem can be difficult to examine. I suggest reading:

https://homepage.univie.ac.at/Dietrich.Burde/papers/burde_39_max_ab.pdf

and

https://link.springer.com/article/10.1007/s00607-009-0029-8

ADDED after question was edited

Now that you have a specific case of course you can say much much more. Since you have $[\mathfrak g,\mathfrak g]=\mathfrak g$ your real Lie algebra is one of the few 8-dimensional semisimple Lie algebras and yes, you have an algorithm (basically computing the signature of the Killing form) to identify it exactly.

Should it be exactly $\mathfrak{sl}(3;\mathbb R)$ (which is the first obvious guess) then you can find a list of isomorphism classes of its subalgebras, e.g. in P. Winternitz. Subalgebras of Lie algebras. Example of sl(3,R). In P. Winternitz, J. Harnad, C.S. Lam, and J. Patera, editors, Symmetry in Physics. In memory of Robert T. Sharp, volume 34 of CRM Proceedings and Lecture Notes, pages 215–227, Montreal, QC, Canada, 2004. AMS, Providence, R.I.

In general I think you should look for the work of Winternitz, Patera and collaborators which wrote much on this sort of specific examples, usually directly derived from the study of symmetry conditions on specific differential equations.

There are some known, quite explicit, algorithms for the case in which $\mathfrak h$ is abelian. The fact that this case is already quite difficult shows how the problem can be difficult to examine. I suggest reading:

https://homepage.univie.ac.at/Dietrich.Burde/papers/burde_39_max_ab.pdf

and

https://link.springer.com/article/10.1007/s00607-009-0029-8

ADDED after question was edited

Now that you have a specific case of course you can say much much more. Since you have $[\mathfrak g,\mathfrak g]=\mathfrak g$ your real Lie algebra is one of the few 8-dimensional semisimple Lie algebras and yes, you have an algorithm (basically computing the signature of the Killing form) to identify it exactly.

Should it be exactly $\mathfrak{sl}(3;\mathbb R)$(** No it is not: see José's answer) (which is the first obvious guess) then you can find a list of isomorphism classes of its subalgebras, e.g. in P. Winternitz. Subalgebras of Lie algebras. Example of sl(3,R). In P. Winternitz, J. Harnad, C.S. Lam, and J. Patera, editors, Symmetry in Physics. In memory of Robert T. Sharp, volume 34 of CRM Proceedings and Lecture Notes, pages 215–227, Montreal, QC, Canada, 2004. AMS, Providence, R.I.

In general I think you should look for the work of Winternitz, Patera and collaborators which wrote much on this sort of specific examples, usually directly derived from the study of symmetry conditions on specific differential equations.

There are some known, quite explicit, algorithms for the case in which $\mathfrak h$ is abelian. The fact that this case is already quite difficult shows how the problem can be difficult to examine. I suggest reading:

https://homepage.univie.ac.at/Dietrich.Burde/papers/burde_39_max_ab.pdf

and

https://link.springer.com/article/10.1007/s00607-009-0029-8

There are some known, quite explicit, algorithms for the case in which $\mathfrak h$ is abelian. The fact that this case is already quite difficult shows how the problem can be difficult to examine. I suggest reading:

https://homepage.univie.ac.at/Dietrich.Burde/papers/burde_39_max_ab.pdf

and

https://link.springer.com/article/10.1007/s00607-009-0029-8

ADDED after question was edited

Now that you have a specific case of course you can say much much more. Since you have $[\mathfrak g,\mathfrak g]=\mathfrak g$ your real Lie algebra is one of the few 8-dimensional semisimple Lie algebras and yes, you have an algorithm (basically computing the signature of the Killing form) to identify it exactly.

Should it be exactly $\mathfrak{sl}(3;\mathbb R)$ (which is the first obvious guess) then you can find a list of isomorphism classes of its subalgebras, e.g. in P. Winternitz. Subalgebras of Lie algebras. Example of sl(3,R). In P. Winternitz, J. Harnad, C.S. Lam, and J. Patera, editors, Symmetry in Physics. In memory of Robert T. Sharp, volume 34 of CRM Proceedings and Lecture Notes, pages 215–227, Montreal, QC, Canada, 2004. AMS, Providence, R.I.

In general I think you should look for the work of Winternitz, Patera and collaborators which wrote much on this sort of specific examples, usually directly derived from the study of symmetry conditions on specific differential equations.

I fail to see how the Levi decomposition of the $\mathfrak g$ should be related to the Levi decomposition of $\mathfrak h$. By identifying $\mathfrak h$ with a Lie subalgebra in $\mathfrak g$ it is clear that the Levi decomposition of the whole algebra does not bring much information on the LEvi decomposition of the subalgebra (as the example of $\mathfrak g=\mathfrak{gl}_n$ shows...)

There are some known, quite explicit, algorithms for the case in which $\mathfrak h$ is abelian. The fact that this case is already quite difficult shows how the problem can be difficult to examine. I suggest reading:

https://homepage.univie.ac.at/Dietrich.Burde/papers/burde_39_max_ab.pdf

and

https://link.springer.com/article/10.1007/s00607-009-0029-8