According to wikipedia, an irreducible Riemannian symmetric space is a Riemannian symmetric space that cannot be written as a product of Riemannian symmetric spaces. I am trying to figure out whether $SL(n,\mathbb{C})/SU(n)$ is an irreducible Riemannian symmetric space. I see that these spaces are completely classified but I couldn't find it.
If it is irreducible, can anyone provide a reference where it is shown (or an argument explaining why it is true) ? And if not, what would be the product decomposition ?
Thank you for your help.