How do I get the columns that a statsmodels / patsy formula depends on?

Suppose I have a pandas dataframe:

df = pd.DataFrame({'x1': [0, 1, 2, 3, 4], 
                   'x2': [10, 9, 8, 7, 6], 
                   'x3': [.1, .1, .2, 4, 8], 
                   'y': [17, 18, 19, 20, 21]})

Now I fit a statsmodels model using a formula (which uses patsy under the hood):

import statsmodels.formula.api as smf
fit = smf.ols(formula='y ~ x1:x2', data=df).fit()

What I want is a list of the columns of df that fit depends on, so that I can use fit.predict() on another dataset. If I try list(fit.params.index), for example, I get:

['Intercept', 'x1:x2']

I've tried recreating the patsy design matrix, and using design_info, but I still only ever get x1:x2. What I want is:

['x1', 'x2']

Or even:

['Intercept', 'x1', 'x2']

How can I get this from just the fit object?

Simply test if the column names appear in the string representation of the formula:

ols = smf.ols(formula='y ~ x1:x2', data=df)
fit = ols.fit()

print([c for c in df.columns if c in ols.formula])
['x1', 'x2', 'y']

There is another approach by reconstructing the patsy model (more verbose, but also more reliable) and it does not depend on the original data frame:

md = patsy.ModelDesc.from_formula(ols.formula)
termlist = md.rhs_termlist + md.lhs_termlist

factors = []
for term in termlist:
    for factor in term.factors:
        factors.append(factor.name())

print(factors)
['x1', 'x2', 'y']