Is there any way of embedding $\mathfrak{so}(n)$ into $\mathfrak{su}(n)$ for any $n$ other than picking the antisymmetric matrices of $\mathfrak{su}(n)$? I know that for small $n$ one can use isomorphisms to do this, but I'm wondering if it is possible to do this for a general $n$.
In other words, is there a non-standard way of representing $\mathfrak{so}(n)$ using $\mathfrak{su}(n)$ matrices?
cheers!
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$\begingroup$Yes, we can embed the groups $SO(n)\hookrightarrow SU(n)$, and also the Lie algebras $\mathfrak{so}(n)\hookrightarrow \mathfrak{su}(n)$ for all $n$. This is a classical case arising in nuclear physics and in the theory of Riemannian symmetric spaces. For references see the section Important subgroups here, where $SO(n)\subseteq SU(n)$ is explained (which of course implies $\mathfrak{so}(n)\subseteq \mathfrak{su}(n)$ for all $n$). Dynkin has classified all possibilities of embeddings among classical Lie algebras.
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