A general Lie triple system as defined by K. Yamaguti, is considered as an anticommutative algebra
$A$ with a trilinear operation
$\left[x,y,z\right]$ in which (among other
things) the mappings
$D\left(x,y\right):z\to \left[x,y,z\right]$
are derivations of
$A$.
It is shown that if the trilinear operation is homogeneous, and
$A$
is irreducible as a general L.t.s. or irreducible relative to the Lie algebra
$I\left(A\right)$ generated
by the
$D\left(x,y\right)$’s,
then
$A$
is a simple algebra. The main result is the following. If
$A$ is a
simple finitedimensional anticommutative algebra over a field of characteristic
zero which is a general L.t.s. with a homogeneous trilinear operation
$\left[x,y,z\right]$, then
$A$
is (1) a Lie algebra; or (2) a Malcev algebra; or (3) an algebra satisfying
$J\left(x,y,z\right)w=J\left(w,x,yz\right)+J\left(w,y,zx\right)+J\left(w,z,xy\right)$ where
$J\left(x,y,z\right)=xy\cdot z+yz\cdot x+zx\cdot y$. Furthermore in all
three cases
$I\left(A\right)$ is the
derivation algebra of
$A$
and
$I\left(A\right)$ is completely
reducible in
$A$.
