How to Construct Gamma Matrices in Higher Dimensions Using Sigma Matrices?

[y35dp]

Homework Statement

If D =7 and the metric g^\mu\nu=diag(+------), Using the outer product of matrices, A \otimes B construct a suitable set of \gamma matrices from the 2 x 2 \sigma-matrices

Homework Equations

\sigma¹=(0, 1 ) \sigma²=(0, -i)
(1, 0) (i, 0)
\o³=(1, 0)
(0, -1)
we need only refer to the basic properties of the sigma matrices

\sigma_i\sigma_j = i \epsilon_ijk\sigma_k + \delta_ijI₂

and

\sigma₁^T=\sigma₁, \sigma₂^T=\sigma₂, \sigma₃^T=-\sigma₃, \sigma₁^*=\sigma₁, \sigma₂^*=\sigma₃^*=-\sigma₃

The Attempt at a Solution

As of yet I have found no \gamma-matrices that satisfy {\gamma^\mu, \gamma^\nu} = 2g^\mu\nu. The closest I have come is a set which satisfy {\gamma^\mu, \gamma^\nu} = 2\delta_\mu\nuI₇

 

[y35dp]

Sorry those \sigma¹,\sigma²,\sigma³ are supposed to be the Pauli matrices, pretty poor attempt at making matrices on my part

 

[fzero]

If you have a matrix that satisfies M^2 = I, there's always a matrix N that's a scalar multiple of M such that N^2 = -I.