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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.

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