What does $\mathrm{sl}_1(B)$ mean when $B$ is a quaternion algebra over a field?

Question

I encountered this in a paper, and I don't know what it actually means.

Let $K$ be a finite extension of $\mathbf Q_p$, $B$ a central simple algebra of degree $2$ over $K$, and consider the Lie algebra $\mathrm{sl}_1(B)$.

(I realise Lie algebras are usually written using \mathfrak but the paper uses \mathrm so I'm following suit.)

Now if $B$ were a field then I would simply interpret $\mathrm{sl}_1(B)$ as $\{0\}$, as that would be the only element of $B$ with trace $0$, but clearly that's not what the author means.

Answer 1

The norm one group of $B$ is defined by $$ G:=SL_1(B):=\lbrace x\in B\mid Nrd_{B/F}(x)=1\rbrace . $$ This is an algebraic group, and we denote its Lie algebra by $sl_1(B)$. (For a definition over $p$-adic fields see for example section $3$ here).