Preheat oven to 350 F with rack in middle. Generously butter pan and dust with flour, knocking out excess.
Whisk together flour, cardamom, baking powder, baking soda, and salt. Beat together butter and granulated sugar in mixer at medium speed, scraping side of bowl occasionally, until pale and fluffy, about 5 minutes. Scrape seeds from vanilla bean with tip of paring knife into butter mixture, reserving pods for another use, and beat until combined well, about 1 minute. Add eggs 1 at a time, beating well after each addition, then beat in lemon juice until combined well. At low speed, add flour mixture and milk alternately in batches, beginning and ending with flour mixture, mixing until just combined.
Spoon batter into pan, smoothing top. Gently rap pan on counter to eliminate air bubbles.
Bake until a wooden skewer inserted into center of cake comes out clean, about 1 hour. Cool in pan 1 hour, then invert onto a rack and cool completely, about 1 hour more.
For whipping cream: Beat cream with confectioners sugar and vanilla extract using whisk attachment of mixer until it just holds stiff peaks. Serve cake with whipped vanilla cream.