The double Capelli polynomial of total degree
2
t
2t
is
∑
{
(
s
g
σ
τ
)
x
σ
(
1
)
y
τ
(
1
)
x
σ
(
2
)
y
τ
(
2
)
⋯
x
σ
(
t
)
y
τ
(
t
)
|
σ
,
τ
∈
S
t
}
.
\begin{equation*} \sum \left \{ (\mathrm {sg}\, \sigma \tau ) x_{\sigma (1)}y_{\tau (1)}x_{\sigma (2)}y_{\tau (2)}\cdots x_{\sigma (t)}y_{\tau (t)} |\; \sigma ,\, \tau \in S_t\right \}. \end{equation*}
It was proved by Giambruno-Sehgal and Chang that the double Capelli polynomial of total degree
4
n
4n
is a polynomial identity for
M
n
(
F
)
M_n(F)
. (Here,
F
F
is a field and
M
n
(
F
)
M_n(F)
is the algebra of
n
×
n
n \times n
matrices over
F
F
.) Using a strengthened version of this result obtained by Domokos, we show that the double Capelli polynomial of total degree
4
n
−
2
4n-2
is a polynomial identity for any proper
F
F
-subalgebra of
M
n
(
F
)
M_n(F)
. Subsequently, we present a similar result for nonsplit inequivalent extensions of full matrix algebras.