Let $\mathfrak{g}$ be a Lie algebra and let $\mathfrak{h}$ be a subalgebra. According to wikipedia, $\mathfrak{h}$ is called an ideal of $\mathfrak{g}$ if it satisfies the condition that $$[\mathfrak{h},\mathfrak{g}]\subseteq\mathfrak{h}\text{.}$$
Occasionally I see a notion called "a two-sided Lie ideal" of a Lie algebra (See here for an example). How is it defined? Is it the same as an ideal as is defined above?
A one sided ideal in a Lie algebra is also a two sided ideal. This is because an ideal is a subspace and hence if $[h,g]\in \mathfrak h$ we also have $[g,h]=-[h,g]\in \mathfrak h$.