Let g be a complex Lie algebra, of dimension greater than one. Does there always exist a subalgebra of dimension dim(g)-1? If the claim is not true, is it true for some particular dimension? I can see that the claim is trivially true for dim(g)=2, but what about for example dim(g)=3? The o(3,R) spanned by e_1,e_2,e_3, (with real coefficients) satisfying [e_1,e_2]=e_3, [e_2,e_3]=e_1, [e_3,e_1]=e_2, doesn't have a two dimensional subalgebra, but the obvious counter example attempt o(3,C) (the same basis, but complex coefficients) to the claim doesn't work, because now e_1+ie_2 and e_3 actually span a two dimensional subalgebra, [e_1+ie_2,e_3] = i(e_1 + ie_2).