So far I haven't found a source that gives me the unambiguous definition of a Lie algebra being real. So: do we call a Lie algebra real if it has real structure constants OR, when viewed as a vector space, the field over this vector space is $\mathbb{R}$. For example, the algebra consisting of $\{L_1,L_2,L_3\}$ with $[L_i,L_j]=i\epsilon_{ijk}L_k$: is $\text{span}_{\mathbb{R}}\{L_1,L_2,L_3\}$ a real Lie algebra?