Calculate the area of the base triangle ABC.

The base triangle ABC is an equilateral triangle with side length $a = 2\sqrt{3}$.
The formula for the area of an equilateral triangle is $\text{Area} = \frac{\sqrt{3}}{4} a^2$.

$$\text{Area}_{ABC} = \frac{\sqrt{3}}{4} (2\sqrt{3})^2$$ $$\text{Area}_{ABC} = \frac{\sqrt{3}}{4} (4 \times 3)$$ $$\text{Area}_{ABC} = \frac{\sqrt{3}}{4} (12)$$ $$\text{Area}_{ABC} = 3\sqrt{3}$$

The height of the pyramid is $SO = 4$.
The formula for the volume of a pyramid is $V = \frac{1}{3} \times \text{Area of Base} \times \text{Height}$.

$$V = \frac{1}{3} \times \text{Area}_{ABC} \times SO$$ $$V = \frac{1}{3} \times (3\sqrt{3}) \times 4$$ $$V = \sqrt{3} \times 4$$ $$V = \boxed{4\sqrt{3}}$$