group theory - Can Lie algebra $sl(2,mathbb{C})$ be decomposed to direct sum of two $sl(2,mathbb{R})$?

The number of generators of Lie algebra $sl(2,\mathbb{C})$ is 6, and $sl(2,\mathbb{R})$ has 3 generators, Can Lie algebra $sl(2,\mathbb{C})$ be decomposed to direct sum of two $sl(2,\mathbb{R})$? Say \begin{equation} sl(2,\mathbb{C})=sl(2,\mathbb{R}) \oplus sl(2,\mathbb{R}) ~? \tag{1} \end{equation} If this holds, can you give one explicit representation of those generators?

By the way there is a similar relation which I know is hold \begin{equation} so(4)=su(2) \oplus su(2). \tag{2} \end{equation}