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

$$\sigma$$¹=(0, 1 ) $$\sigma$$²=(0, -i)
(1, 0) (i, 0)
$$\o$$³=(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

$$\sigma$$₁^T=$$\sigma$$₁, $$\sigma$$₂^T=$$\sigma$$₂, $$\sigma$$₃^T=-$$\sigma$$₃, $$\sigma$$₁^*=$$\sigma$$₁, $$\sigma$$₂^*=$$\sigma$$₃^*=-$$\sigma₃$$

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$$\delta$$_{$$\mu$$$$\nu$$}I₇

 

[y35dp]

Sorry those $$\sigma$$¹,$$\sigma$$²,$$\sigma$$³ 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 $$N^2 = -I$$.