Is the statement below false?

"The metaplectic group Mp2(R) is not a matrix group: it has no faithful finite-dimensional representations."

Possible "counterexample": Sp(2n,R) is a subgroup of O(4n,C) (or O(2n,2n) if you prefer). So the Clifford algebraic Pin group will contain a double cover. The double cover will definitely be disconnected if Sp(2n,R) is not a subgroup of SO(4n,C). It should be connected if it is entirely in the Spin subgroup of the Pin group.

Consider the case of Sp(2,R). If we have a 2x2 real matrix with determinant 1, we can establish an isomorphism in SO(4,C) as follows: a^2-b^2-c^2+d^2 = 1

[a+b,c-d;c+d,a-b] <---> [a,-bi,-ci,-d;bi,a,d,-ci;ci,-d,a,bi;d,ci,-bi,a]

Since Spin(4,C) will double cover SO(4,C), we could have a connected double cover of Sp(2,R).

Note: The proposed "example" is false due to submitted answer. Thanks.