The commuting scheme of ${\mathfrak{g}}_2$

We study the commuting scheme of the exceptional Lie algebra ${\mathfrak{g}}_2$ via computational methods. Over a field of characteristic zero, we show that the commuting scheme of ${\mathfrak{g}}_2$ is reduced, Cohen-Macaulay and normal. Our computations give an explicit homogeneous system of parameters, determine the Hilbert series and degree, identify the first syzygies of the commuting ideal and show that it is not Gorenstein. We extend some of the above results to positive characteristic.