A question on notation

I'm having a bit of difficulty trying to combine these four equations into one "covariant" type equation:

$\sigma^0 \sigma^0 = \sigma^0$

$\sigma^0 \sigma^i = \sigma^i \;\;;\;\; i = 1,2,3$

$\sigma^i \sigma^0 = \sigma^i \;\;;\;\; i = 1,2,3$

$\sigma^i \sigma^j = \delta^{ij} \sigma^0 + i \epsilon^{ijk} \sigma^k \;\;;\;\; i = 1,2,3\;\; j =1,2,3\;\; k = 1,2,3$

I want an equation in the form:

$\sigma^{\mu} \sigma^{\nu} = ?$
where $\mu,\nu = 0,1,2,3$

The difficultly comes when you spot that $\delta^{\mu \nu} \sigma^0$ fits a good part of the problem but then when you add in expressions like $\delta^{0\mu} \sigma^{\nu}$ it is spoilt when $\nu = 0$ as you start to over count.

Any ideas or answers?

For those interested, the sigmas are the pauli matrices inc. 0 which is the 2x2 identity matrix.

That issue can be dealt with by multiplying by $(1 - \delta^{0 \nu})$

To be honest the expression is going to be so messy that I'm not sure what use it will be, but for what it's worth here's what I get:

$\sigma^{\mu} \sigma^{\nu} = \delta^{\mu \nu}\sigma^{0} + (1 - \delta^{0 \mu})(1 - \delta^{0 \nu}) i \epsilon^{\mu \nu \rho} \sigma^{\rho} + \delta^{0 \mu}(1 - \delta^{0 \nu})\sigma^{\nu} + \delta^{0 \nu}(1 - \delta^{0 \mu})\sigma^{\mu}$