Notational question on Cartan Decomposition of Semisimple Lie Algebra

Question

I've often wondered why in every book of Lie algebra the Cartan Decomposition in written as $$\mathfrak{g}=\mathfrak{h}\oplus\underset{{\scriptscriptstyle \lambda\in\Phi}}{\bigoplus}\mathfrak{g}_{\lambda},$$ instead of $$\mathfrak{g}=\mathfrak{h} \underset{{\scriptscriptstyle \lambda\in\Phi}}{\bigoplus}\mathfrak{g}_{\lambda}.$$ Is it just a notational convention or there's some deeper reason that I'm missing for this notation? Thank you in advance

Answer 1

I'm pretty sure this is just an expansion of Ben West's comment:

If we write $\Phi=\{\lambda_1,\ldots,\lambda_d\}$, then $$\bigoplus_{\lambda\in\Phi}\mathfrak{g}_\lambda=\mathfrak{g}_{\lambda_1}\oplus\cdots\oplus \mathfrak{g}_{\lambda_d}$$ so $$\mathfrak{h}\bigoplus_{\lambda\in\Phi}\mathfrak{g}_\lambda=\mathfrak{h}\mathfrak{g}_{\lambda_1}\oplus\cdots\oplus \mathfrak{g}_{\lambda_d}$$ which doesn't make any sense.