Why a semisimple Lie algebra is always isomorphic to a subalgebra of special linear algebra?

Question

According to wikipedia, every semisimple Lie algebra is a subalgebra of the special linear Lie algebra $\mathfrak{sl}_n$ for some $n\ge 1$. But I do not know any proof of this, nor am I able to imagine an intuitive reason for this remarkable fact. Can you give me a hint or an indication on how to demonstrate this?

Answer 1

Let $\mathfrak g$ be a semisimple Lie algebra over a field $\Bbbk$ with characteristic $0$. According to Ado's theorem, $\mathfrak g$ is isomorphic to a subalgebra $\mathfrak s$ of $\mathfrak{gl}_n$, for some $n\in\Bbb N$. But $\mathfrak{gl_n}$ is isomorphic to a subalgebra of $\mathfrak{sl}_{n+1}$: it's the subalgebra which consits of matrices of the form $\left[\begin{smallmatrix}M&0\\0&-\operatorname{tr}(M)\end{smallmatrix}\right]$, with $M\in\mathfrak{gl}_n$.

Answer 2

Since a semisimple Lie algebra $L$ over any field $K$ has trivial center, because the solvable radical is zero, the adjoint representation is faithful. So we obtain an embedding of Lie algebras $$ L\hookrightarrow \mathfrak{gl}_n(K)\hookrightarrow\mathfrak{sl}_{n+1}(K), $$ without refering to Ado's Theorem, or Iwasawa's Theorem for prime characteristic.