Corollary 1

Let $\mathfrak{g}$ and $\mathfrak{g}'$ be two compact real Lie algebras. Then $\mathfrak{g}$ and $\mathfrak{g}'$ are isomorphic if and only if the complex Lie algebras $\mathfrak{g}_{(\mathbf{C})}$ and $\mathfrak{g}'_{(\mathbf{C})}$ are isomorphic.

The condition is clearly necessary. Conversely, assume that $\mathfrak{g}_{(\mathbf{C})}$ and $\mathfrak{g}'_{(\mathbf{C})}$ are isomorphic. Let $\mathfrak{c}$ (resp. $\mathfrak{c}'$) be the centre of $\mathfrak{g}$ (resp. $\mathfrak{g}'$) and $\mathfrak{s}$ (resp. $\mathfrak{s}'$) the derived algebra of $\mathfrak{g}$ (resp. $\mathfrak{g}'$). Then $\mathfrak{c}_{(\mathbf{C})}$ and $\mathfrak{c}'_{(\mathbf{C})}$ are the centres of $\mathfrak{g}_{(\mathbf{C})}$ and $\mathfrak{g}'_{(\mathbf{C})}$, respectively, and hence are isomorphic; it follows that the commutative algebras $\mathfrak{c}$ and $\mathfrak{c}'$ are isomorphic. Similarly, $\mathfrak{s}_{(\mathbf{C})}$ and $\mathfrak{s}'_{(\mathbf{C})}$ are isomorphic, hence $\mathfrak{s}$ and $\mathfrak{s}'$, which are compact real forms of two isomorphic complex semi-simple Lie algebras, are isomorphic by Th. $1b)$.