Is the Lie algebra so(2,6) semi-simple?

Question

I'd like to know whether the Lie algebra so(2,6) is semi-simple. I know that I could compute the Killing form for that. But is there an easier way?

Answer 1

Hint 1: What is the complexification of that Lie algebra?

Hint 2: Do you know that for all $n \ge 3$, $\mathfrak{so}(n, \mathbb C)$ is semisimple (actually simple unless $n=4$)?

Hint 3: What do you know about the relation of semisimplicity of a Lie algebra to the semisimplicity of its complexification?