There are actually four statistics that are used for MANOVA: Pillai's trace, Hotelling-Lawley's trace, Wilks' lambda, and Roy's largest root. In general, the four of them will differ, and Pillai's trace is considered to be the best. All of these tests work with the sum of squares (we have two sum of squares terms: the model one and the error one). In our case, these two will be 3 x 3 matrices and, under the null hypothesis, the means are the same (no group effect); in consequence, the model sum of squares should be close to the error one. If that is not the case, and the model sum of squares is larger than the error one, we may conclude that the group effect is significative:

SS = summary(result)$SS
print(SS)

The preceding code generate the following output: 

In particular, Pillai's test uses the following statistic (where H is the model SS and E is the error SS):

The idea is that under the null hypothesis, H will be roughly similar to H + E, because H should be small (the sum of squares explained by the model). The other tests use different test statistics, but the general idea remains the same.