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Consider the Lie group ${\rm Sp}(6,\mathbb{R})$ defined as

${\rm Sp}(6,\mathbb{R}) = \{g\in {\rm GL}(6,\mathbb{R}): {}^tgJg=J \},$

where

$J=\begin{bmatrix} &&&&&1\\ &&&&1&\\ &&&1&&\\ &&-1&&&\\ &-1&&&&\\ -1&&&&&\\ \end{bmatrix}.$

Its Lie algebra $sp(6,\mathbb{R})$ is the set of all $6\times 6$ matrices $X$ with real entries such that

${}^tXJ+JX=0.$

A general element of $sp(6,\mathbb{R})$ is of the form

$\begin{bmatrix} a & b & c & d & e & f\\ g & h & i & j & k & e\\ l & m & n & o & j & d\\ p & q & r & -n & -i & -c\\ s & t & q & -m & -h & -b\\ u & s & p & -l & -g & -a\\ \end{bmatrix}$

and so $sp(6,\mathbb{R})$ is a 21-dimensional real vector space. We would like to describe its root system.