Many thanks! I will delve into that.

Okay, thank you. I will have a look at this.

Yes I see, thank you for the suggestion. This may answer part of the issue. However I am looking for a complete characterization of them being isomorphic or not. Therefore: what if the complexified representations are isomorphic using the above strategy? Would you have an idea? Many thanks again.

Yes thanks, I can do that. But, would you also know a generalization to the real Lie algebra case?

Many thanks for your advice. I finally found a way to get around this computation for my specific problem. But I will definitely keep the name of de Graaf in mind.

@Jim Thank you for your suggestion. I do not have access to this book, but from the "Contents" and the "Index of Algorithms" at the end, I am not sure that it provides a (straight) answer. I have also read other articles of de Graaf and I did not find anything.