  The projective orthogonal and symplectic groups $PO_n(F)$ and $PSp_n(F)$ have
a natural action on the $F$ vector space $V' = M_n(F) \oplus ... \oplus
M_n(F)$. Here we assume $F$ is an infinite field of characteristic not 2. If we
assume there is more than one summand in $V'$, then the invariant fields
$F(V')^{PO_n}$ and $F(V')^{PSp_n}$ are natural objects. They are, for example,
the centers of generic algebras with the appropriate kind of involution. This
paper considers the rationality properties of these fields, in the case $1,2$
or 4 are the highest powers of 2 that divide $n$. We derive rationality when
$n$ is odd, or when 2 is the highest power, and stable rationality when 4 is
the highest power. In a companion paper [ST] joint with Tignol, we prove
retract rationality when 8 is the highest power of 2 dividing $n$. Back in this
paper, along the way, we consider two generic ways of forcing a Brauer class to
be in the image of restriction.
