Suppose that L = L1 ⊕ L2 is the direct sum of two Lie algebras. Now suppose that L1 and L2 do not have any non-trivial proper ideals. Let J be a proper ideal of L. Show that if J ∩ L1 = 0 andJ ∩ L2 = 0, then the projections p1 : J → L1 and p2 : J → L2 are isomorphisms.
In order to show these two are isomorphisms I'm trying to show injection and surjection. I've shown injection through the fact that ker(p1)=0=ker(p2), but I'm still struggling with the surjection part of the proof. Any tips would be appreciated.
Show that the images of those projections are ideals in the respective $L_i$, so by assumption they are either everything or zero. Now if e.g. $im(p_1) = \lbrace 0\rbrace$, it follows that $J\subseteq L_2$ and hence $J=J\cap L_2 =\lbrace 0\rbrace$.