If $mathfrak{g}=bigoplusmathfrak{g}_i$ is a semisimple Lie algebra, why does...
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There is this property about Cartan subalgebras that is not clear to me. Suppose $mathfrak{g}$ is a semisimple Lie algebra. Then I know we can decompose it uniquely as $mathfrak{g}=bigoplusmathfrak{g}_i$, where the $mathfrak{g}_i$ are simple ideals. If $mathfrak{h}$ is a Cartan subalgebra (maximal toral subalgebra), then $mathfrak{h}=bigoplus(mathfrak{h}capmathfrak{g}_i)$. The $supseteq$ direction is clear. Why does the other follow? I thought of something like this, take $xinmathfrak{h}$, and write $x=sum x_i$ for $x_iinmathfrak{g}_i$. I want to show $x_iinmathfrak{h}$ for each $i$. I did this by induction on the number of nonzero summands. If there is only one summand so $x=x_i$, the claim is clear. If there's more than one summand, I'm not sure how to reduce it to imply the induction...