I've seen it claimed in physics classes that there is an isomorphism of Lie algebras $$\mathfrak{so}\left(1,3\right)\cong \mathfrak{su}\left(2\right)\oplus\mathfrak{su}\left(2\right)$$ However, the argument used relied on taking complex linear combinations of a basis for $\mathfrak{so}\left(1,3\right)$ so that the resulting basis contained two copies of $\mathfrak{su}\left(2\right)$.
Namely, there is a basis for $\mathfrak{so}\left(1,3\right)$ given by $\left\{J_1,J_2,J_3,K_1,K_2,K_3\right\}$ with $$\left[J_i,J_j\right]=\sum _{k=1}^3\varepsilon _{ijk}J_k$$ $$\left[K_i,K_j\right]=\sum _{k=1}^3\varepsilon _{ijk}J_k$$ $$\left[J_i,K_j\right]=\sum _{k=1}^3\varepsilon _{ijk}K_k$$ The new elements $J_i^{\pm}=J_i\pm iK_i$ then have the commutation relations of two copies of $\mathfrak{su}\left(2\right)$.
The problem I have with this derivation is that $\mathfrak{so}\left(1,3\right)$ and $\mathfrak{su}\left(2\right)$ are real Lie algebras. So, taking complex linear combinations doesn't really make sense.
Is the stated result true?
$\mathfrak{so}(1, 3)$, while a real Lie algebra, happens to be isomorphic as a real Lie algebra to the complex Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. This Lie algebra is the complexification of the real Lie algebra $\mathfrak{su}(2)$, so we have a direct sum decomposition
$$\mathfrak{sl}_2(\mathbb{C}) \cong \mathfrak{su}(2) \oplus i \mathfrak{su}(2)$$
but this is not a Lie algebra direct sum; the factor $i \mathfrak{su}(2)$ isn't closed under Lie bracket. It is just a way of expressing that $\mathfrak{sl}_2(\mathbb{C})$ is the complexification of $\mathfrak{su}(2)$.