Notes on "Einstein metrics on compact simple Lie groups attached to standard triples"
  In the paper "Einstein metrics on compact simple Lie groups attached to
standard triples", the authors introduced the definition of standard triples
and proved that every compact simple Lie group $G$ attached to a standard
triple $(G,K,H)$ admits a left-invariant Einstein metric which is not naturally
reductive except the standard triple $(\Sp(4),2\Sp(2),4\Sp(1))$. For the triple
$(\Sp(4),2\Sp(2),4\Sp(1))$, we find there exists an involution pair of $\sp(4)$
such that $4\sp(1)$ is the fixed point of the pair, and then give the
decomposition of $\sp(4)$ as a direct sum of irreducible
$\ad(4\sp(1))$-modules. But $\Sp(4)/4\Sp(1)$ is not a generalized Wallach
space. Furthermore we give left-invariant Einstein metrics on $\Sp(4)$ which
are non-naturally reductive and $\Ad(4\Sp(1))$-invariant. For the general case
$(\Sp(2n_1n_2),2\Sp(n_1n_2),2n_2\Sp(n_1))$, there exist $2n_2-1$ involutions of
$\sp(2n_1n_2)$ such that $2n_2\sp(n_1))$ is the fixed point of these $2n_2-1$
involutions, and it follows the decomposition of $\sp(2n_1n_2)$ as a direct sum
of irreducible $\ad(2n_2\sp(n_1))$-modules. In order to give new non-naturally
reductive and $\Ad(2n_2\Sp(n_1)))$-invariant Einstein metrics on
$\Sp(2n_1n_2)$, we prove a general result, i.e. $\Sp(2k+l)$ admits at least two
non-naturally reductive Einstein metrics which are
$\Ad(\Sp(k)\times\Sp(k)\times\Sp(l))$-invariant if $k<l$. It implies that every
compact simple Lie group $\Sp(n)$ for $n\geq 4$ admits at least
$2[\frac{n-1}{3}]$ non-naturally reductive left-invariant Einstein metrics.
