= D5 root system
{c}
{title2=$D_5$}
The <root system> $D_5$ consists of the forty <roots> $\pm e_i\pm e_j$, $1\le i<j\le5$, in an orthonormal <basis>. One choice of <simple roots> is $e_1-e_2,e_2-e_3,e_3-e_4,e_4-e_5,e_4+e_5$. The <Dynkin diagram> has a fork at node $3$ with ends $4$ and $5$. Its <Weyl group> permutes coordinates and changes an even number of signs. The associated complex <simple Lie algebra> is $\mathfrak{so}(10,\mathbb C)$. Its two <half-spin representations> have dimension $16$, with <highest weights> $\tfrac12(e_1+e_2+e_3+e_4\mp e_5)$.
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