This paper deals with a class of triple systems satisfying two generalized five linear identities and having nondegenerate bilinear forms with certain properties. If
(
M
,
{
,
,
}
)
(M,\{ ,,\} )
is such a triple system with bilinear form
ϕ
(
,
)
\phi (,)
, it is shown that if
M
M
is semisimple, then
M
M
is the direct sum of simple ideals if
ϕ
\phi
is symmetric or symplectic or if
M
M
is completely reducible as a module for its right multiplication algebra
L
\mathcal {L}
. It is also shown that if
M
M
is a completely reducible
L
\mathcal {L}
-module,
M
M
is the direct sum of a semisimple ideal and the center of
M
M
. Such triple systems can be embedded into certain nonassociative algebras and the results on the triple systems are extended to these algebras.