In this paper, we study algebras with involution that are isomorphic after base field extension to the tensor product of two composition algebras. To any such algebra
(
A
,
−
)
(\mathcal {A},\, - )
, we associate a quadratic form
Q
Q
called the Albert form of
(
A
,
−
)
(\mathcal {A},\, - )
. The Albert form is used to give necessary and sufficient conditions for two such algebras to be isotopic. Using a Lie algebra construction of Kantor, we are then able to give a description of the isomorphism classes of Lie algebras of index
F
4
,
1
21
F_{4,1}^{21}
,
2
E
6
,
1
29
{}^2E_{6,1}^{29}
,
E
7
,
1
48
E_{7,1}^{48}
and
E
8
,
1
91
E_{8,1}^{91}
. That description is used to obtain a classification of the indicated Lie algebras over
R
(
(
T
1
,
…
,
T
n
)
)
,
n
⩽
3
{\mathbf {R}}(({T_1}, \ldots ,{T_n})),\;n \leqslant 3
.