Direct sum of Vector space and Lie Algebra

Question

I frequently encounter the following situation:

One has a Lie algebra, with an underlying vectorspace. Now one time one wants show that $g=h \oplus i$ so $h$ and $i$ are commuting subalgebras (direct sum of Lie algebras) and one time as vectorspaces without the restriction that they are commuting.

I have seen for example in "Barut,Rączka - Theory of Group Representations and Applications" the following notation: $g= h \dot + i$ for the vector space direct sum and $g=h \oplus i$ for the direct sum of Lie algebras.

I have also encountered $g=h \oplus i$ for the vector space direct sum and $g=h \times i$ for the Lie algebra direct sum.

Is there a commonly accepted notation or a standard way to handle this?

Answer 1

All notations you have mentioned are used. The notation $\mathfrak{g}=\mathfrak{a}\oplus \mathfrak{b}$ is often commented, such as "we consider the direct Lie algebra sum", or "we consider the direct vector space sum". Then everything is clear.